Browse Coordinate geometry

$C$ has equation $y = \text{f}(x)$ where $\text{f}(x)$ is a quadratic expression
$C$ has a maximum point at $(1,\ 16)$
$C$ crosses the $x$ -axis at $-3$ and at $k$ , where $k$ is a constant

(b) deduce the value of

k

,

(1 mark)
(c) find

\text{f}(x)

.

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

l

C

and the

y

-axis.
(d) Use inequalities to define

R

.

(2 marks)

●●●●●Level 48 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = \dfrac{x - 1}{x^2 + 3}

.
(a) State the coordinates of the points where the curve crosses the axes.

(2 marks)

The point

A

is where the curve crosses the

y

-axis. The point

B

on the curve has

x

-coordinate

-3

.
(b) Show that the line

AB

is parallel to the

x

-axis.

(2 marks)
(c) Determine the coordinates of the turning points of the curve.

(4 marks)
(d) Determine whether each turning point is a maximum or a minimum.

(5 marks)

●●●●●Level 513 marksStart

The diagram shows the graph of

y = |3x - 6|

.(a) State the coordinates of the points where the graph meets the axes.

(2 marks)
(b) Given that the graphs of

y = |3x - 6|

and

y = ax + 3

have two distinct points of intersection, determine

(i) the set of possible values of

a

,

(ii) the

x

-coordinates of the points of intersection of these graphs, giving your answers in terms of

a

.

(7 marks)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

The line

l_1

passes through the points

P(-3,\ 8)

and

Q(7,\ 3)

, as shown in Figure 1.
(a) Find the equation of

l_1

, giving your answer in the form

y = mx + c

, where

m

and

c

are constants to be found.

(3 marks)

The line

l_2

is perpendicular to

l_1

and passes through the origin

O

.

The point

R

is the point of intersection of

l_1

and

l_2

, as shown in Figure 1.
(b) Show that

R

has coordinates

\left(\tfrac{13}{5},\ \tfrac{26}{5}\right)

(4 marks)
(c) Find the exact area of triangle

OPQ

.

(3 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Given that

the point $A$ has coordinates $(2,\ -1)$
the point $B$ has coordinates $(8,\ 7)$
the line $l_1$ passes through $A$ and $B$

(a) find an equation for

l_1

, giving your answer in the form

px + qy + r = 0

where

p

q

and

r

are integers to be found.

(3 marks)

The line

l_2

passes through

B

and is parallel to the

x

-axis.

The point

C

lies on

l_2

so that the length of

AC

\sqrt{89}

(b) Show that the

x

coordinate of

C

satisfies

x^2 - 4x - 21 = 0

(3 marks)
(c) Hence find both possible pairs of coordinates of the point

C

.

(2 marks)
(d) Hence find the minimum possible area of triangle

ABC

.

(2 marks)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows part of the curve

C

with equation

y = \text{f}(x)

where

\text{f}(x) = 3x^2 - 18x + 32

(a) Write

3x^2 - 18x + 32

in the form

a(x + b)^2 + c

where

a

b

and

c

are constants to be found.

(3 marks)

Given that

C

has a minimum at the point

P

(b) state the coordinates of

P

(1 mark)

The point

Q

lies on

C

and has

x

coordinate

1

The line

l

passes through

Q

and

P

, as shown in Figure 1.
(c) Show that an equation of

l

y = 23 - 6x

(3 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by the

x

-axis, the

y

-axis,

l

and

C

.
(d) Use inequalities to define the region

R

.

(3 marks)

●●●●●Level 510 marksStart

The line

y = mx + 4

and the curve

y = x^{2} - 2x + 8

are drawn on the same axes, where

m

is a constant.
(a) Find the two values of

m

for which the line is a tangent to the curve. (4 marks)
(b) For each of these values of

m

, find the coordinates of the point where the line touches the curve. (3 marks)
(c) Find the set of values of

m

for which the line meets the curve at two distinct points. (2 marks)

●●●●●Level 59 marksStart

The curve

C

has equation

y = \dfrac{x}{\sqrt{2x + 1}}, \qquad x \geqslant 0

The point

A

C

has

x

-coordinate

4

, and

O

is the origin.

The finite region

R

is bounded by

C

and the line segment

OA

.

Using the substitution

u = \sqrt{2x + 1}

, find the exact area of

R

. (9 marks)

●●●●●Level 59 marksStart

In this question you must show detailed reasoning.

The curve

C

has equation

x^2 - 2xy + 2y^2 = 8

The tangent to

C

at the point

P

is parallel to the

x

-axis, and the tangent to

C

at the point

Q

is parallel to the

y

-axis. Both

P

and

Q

lie in the first quadrant.

Determine the exact area of triangle

OPQ

, where

O

is the origin. (7 marks)

●●●●●Level 57 marksStart

In this question you must show detailed reasoning.

Figure 1 shows the curve with equation

y = 6x - x^2

and the line with equation

y = 2x

. The region

R

, shown shaded, is bounded by the curve and the line.Figure 1
(a) Find the coordinates of the two points where the line meets the curve. (3 marks)
(b) Find the area of

R

. (4 marks)
(c) The line with equation

y = kx

, where

0 < k < 6

, divides

R

into two regions of equal area.

Show that

(6 - k)^3 = 32

, and hence find the value of

k

3

significant figures. (3 marks)

●●●●●Level 510 marksStart

Parallel and perpendicular lines

15 questions

LessonNot started

The gradient of the normal to the curve

y = 3x^2 - 2

at the point where

x = 1

is

Circle your answer.

A $6$
B $-6$
C $\dfrac16$
D $-\dfrac16$

●●●●●Level 11 markStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Given that

the point $A$ has coordinates $(5\sqrt{2},\ -1)$
the point $B$ has coordinates $(\sqrt{2},\ 5)$
the line $l_1$ passes through $A$ and $B$

(a) show that

l_1

has gradient

p\sqrt{2}

, where

p

is a rational constant to be found.

You must show each step of your working.

(2 marks)

The line

l_2

passes through

B

and is perpendicular to

l_1

(b) Find an equation for

l_2

, giving your answer in the form

y = mx + c

, where

m

and

c

are constants.

(3 marks)

●●●●●Level 35 marksStart

The points

A(1, 5)

and

B(7, 1)

are given.
(a) Find an equation of the perpendicular bisector of

AB

. (4 marks)
(b) The perpendicular bisector meets the coordinate axes at the points

P

and

Q

. Find the area of triangle

OPQ

, where

O

is the origin. (2 marks)

●●●●●Level 36 marksStart

A triangle has vertices

A(0, 1)

B(6, 3)

and

C(2, 5)

. Show that the triangle is right-angled at

C

●●●●●Level 33 marksStart

The points are

A(-2, 5)

and

B(4, -1)

. The perpendicular bisector of

AB

crosses the

x

-axis at

P

. Find the coordinates of

P

●●●●●Level 34 marksStart

The points are

A(1, 2)

and

B(5, 8)

.
(a) Find the coordinates of the midpoint of

AB

. (1 mark)
(b) Find an equation of the perpendicular bisector of

AB

. (5 marks)

●●●●●Level 36 marksStart

The points

P

and

Q

have coordinates

(-3, 7)

and

(5, 1)

respectively.
(a) Find the coordinates of the midpoint of

PQ

(1 mark)
(b) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(4 marks)

●●●●●Level 35 marksStart

The line

x + 2y = 24

is the normal to the curve

y = a\sqrt{x} + \dfrac{b}{x}

at the point where

x = 4

.

Determine the values of the constants

a

and

b

.

(8 marks)

●●●●●Level 48 marksStart

A circle

C

has equation

x^2 + y^2 - 4x + 6y + k = 0

.
(a) Find the set of possible values of

k

.

(2 marks)

It is given that

k = -7

.
(b) Determine the coordinates of the two points on

C

at which the gradient of the tangent is

2

.

(5 marks)

●●●●●Level 47 marksStart

The points

A

B

and

C

have coordinates

(-2, 6)

(3, 7)

and

(10, 0)

respectively.
(a) Find the equation of the perpendicular bisector of

AB

. Give your answer in the form

y = mx + c

.

(3 marks)
(b) In this question you must show detailed reasoning.

Hence, or otherwise, find an equation of the circle that passes through points

A

B

and

C

.

(5 marks)

●●●●●Level 48 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The line

l_1

has equation

2x + 3y - 26 = 0

The line

l_2

passes through the origin and is perpendicular to

l_1

(a) Find an equation for

l_2

(2 marks)

The lines

l_1

and

l_2

intersect at the point

P

.
(b) Use algebra to find the coordinates of

P

.

(3 marks)
(c) Hence find the shortest distance from the origin to

l_1

, giving your answer as a fully simplified surd.

(2 marks)

●●●●●Level 47 marksStart

Figure 1 shows the circle

C

with equation

x^2 + y^2 - 8x + 2y - 8 = 0

together with the tangent to

C

at the point

P(7, 3)

.Figure 1 (not accurately drawn)

Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers. (5 marks)

●●●●●Level 45 marksStart

The points

P

and

Q

have coordinates

(2, 9)

and

(8, -3)

respectively.
(a) (i) Find the coordinates of the midpoint of

PQ

(ii) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(5 marks)
(b) A circle passes through the points

P

and

Q

The centre of the circle lies on the line with equation

3x - y = 17

(b) (i) Find the equation of the circle.

(ii) The circle intersects the coordinate axes at

n

points.

State the value of

n

(4 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

The line

l_1

passes through the points

P(-3,\ 8)

and

Q(7,\ 3)

, as shown in Figure 1.
(a) Find the equation of

l_1

, giving your answer in the form

y = mx + c

, where

m

and

c

are constants to be found.

(3 marks)

The line

l_2

is perpendicular to

l_1

and passes through the origin

O

.

The point

R

is the point of intersection of

l_1

and

l_2

, as shown in Figure 1.
(b) Show that

R

has coordinates

\left(\tfrac{13}{5},\ \tfrac{26}{5}\right)

(4 marks)
(c) Find the exact area of triangle

OPQ

.

(3 marks)

●●●●●Level 510 marksStart

The lines

L_1

and

L_2

are parallel.

L_1

has equation

3x + 2y = 8

and

L_2

has equation

3x + 2y = 34

L_1

intersects the

y

-axis at the point

P

The point

Q

is the point on

L_2

closest to

P

, as shown in the diagram.(a) (i) Find the coordinates of

Q

(ii) Hence, show that

PQ = k\sqrt{13}

, where

k

is an integer to be found.

(7 marks)
(b) A circle,

C

, has centre

(a, 3)

L_1

and

L_2

are both tangents to

C

(b) (i) Find

a

(ii) Find the equation of

C

(4 marks)

●●●●●Level 511 marksStart

Length and midpoint of a line segment

21 questions

LessonNot started

The distance between

P(1, a)

and

Q(5, 2)

\sqrt{20}

. Find the possible values of

a

●●●●●Level 33 marksStart

The triangle has vertices

A(-1, 2)

B(3, 4)

and

C(5, 0)

. Show that it is isosceles.

●●●●●Level 33 marksStart

The points

P

and

Q

have coordinates

(-3, 7)

and

(5, 1)

respectively.
(a) Find the coordinates of the midpoint of

PQ

(1 mark)
(b) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(4 marks)

●●●●●Level 35 marksStart

A circle

C

has equation

x^2 + y^2 + 12x - 4y = 41

(a) Find

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The point

P

lies on

C

and is the point on

C

furthest away from the origin

O

.
(b) Find the exact length

OP

.

(2 marks)

●●●●●Level 35 marksStart

A circle has equation

x^2 + y^2 - 14x + 6y + 22 = 0

(a) Find

(i) the coordinates of the centre of the circle,

(ii) the radius of the circle.

(3 marks)

Given that

Q

is the point on the circle that is closest to the origin

O

,
(b) find the exact length

OQ

.

(2 marks)

●●●●●Level 35 marksStart

A

is the point

(-1, 3)

and

B

is the point

(h, 9)

.
(a) The gradient of

AB

2

. Find

h

.

(2 marks)
(b)

B

is the midpoint of

AC

. Find the coordinates of

C

.

(2 marks)
(c) The line through

A

B

and

C

meets the curve

y = x^2 - 4x + k

at two distinct points.

Determine the set of possible values of

k

.

(6 marks)

●●●●●Level 410 marksStart

The points

A

B

and

C

have coordinates

(-2, 6)

(3, 7)

and

(10, 0)

respectively.
(a) Find the equation of the perpendicular bisector of

AB

. Give your answer in the form

y = mx + c

.

(3 marks)
(b) In this question you must show detailed reasoning.

Hence, or otherwise, find an equation of the circle that passes through points

A

B

and

C

.

(5 marks)

●●●●●Level 48 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve

C_1

has equation

y = 2x + \frac{10}{x^{2}} \qquad x \neq 0

The curve

C_2

has equation

y = 2x^{2} + 2x - 1

(a) Show that

C_1

and

C_2

meet when

2x^{4} - x^{2} - 10 = 0

(2 marks)

Given that

C_1

and

C_2

meet at the points

P

and

Q

(b) find, using algebra, the exact distance

PQ

(5 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The line

l_1

has equation

2x + 3y - 26 = 0

The line

l_2

passes through the origin and is perpendicular to

l_1

(a) Find an equation for

l_2

(2 marks)

The lines

l_1

and

l_2

intersect at the point

P

.
(b) Use algebra to find the coordinates of

P

.

(3 marks)
(c) Hence find the shortest distance from the origin to

l_1

, giving your answer as a fully simplified surd.

(2 marks)

●●●●●Level 47 marksStart

The line joining the points

(-1, -4)

and

(5, -10)

is the diameter of a circle

C

.
(a) Find an equation for

C

.

(5 marks)
(b) Hence find the exact coordinates of the point on

C

that is nearest the

x

-axis.

(2 marks)

●●●●●Level 47 marksStart

The points

P

and

Q

have coordinates

(2, 9)

and

(8, -3)

respectively.
(a) (i) Find the coordinates of the midpoint of

PQ

(ii) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(5 marks)
(b) A circle passes through the points

P

and

Q

The centre of the circle lies on the line with equation

3x - y = 17

(b) (i) Find the equation of the circle.

(ii) The circle intersects the coordinate axes at

n

points.

State the value of

n

(4 marks)

●●●●●Level 49 marksStart

The circle

C_1

has equation

x^2 + y^2 + 10x - 6y + 9 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(7,\ 8)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 - 12x + 2y + 28 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates

(-2,\ 5)

and radius

k

, where

k

is a positive constant.

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 + 2x - 14y + 34 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(8,\ 19)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

have no points in common,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The diagram shows the circle with equation

x^2 + y^2 - 8y + a = 0

, where

a

is a constant. The centre

C

of the circle lies on the

y

-axis. The line

y = 2x

meets the circle at the points

A

and

B

, and

M

is the midpoint of

AB

.(a) In this question you must show detailed reasoning.

Show that the area of triangle

ABC

\tfrac45\sqrt{64 - 5a}

.

(7 marks)
(b) (i) Find the value of

a

for which this area is zero.

(ii) Give a geometrical interpretation of the case in part (b)(i).

(2 marks)
(c) Give a geometrical interpretation of the case

a = 13

.

(1 mark)

●●●●●Level 510 marksStart

In this question you must show detailed reasoning.The diagram shows the curve with equation

(3x - y)^2 = 12x + 4y

.
(a) Show that

\dfrac{dy}{dx} = \dfrac{9x - 3y - 6}{3x - y + 2}

.

(3 marks)

At the point

P

on the curve the tangent to the curve is parallel to the

y

-axis and at the point

Q

on the curve the tangent to the curve is parallel to the

x

-axis.
(b) Show that the distance

PQ

k\sqrt{10}

, where

k

is a rational number to be determined.

(8 marks)

●●●●●Level 511 marksStart

A curve has parametric equations

x = 2t^2

y = t^3

. The point

P

on the curve has parameter

p

, where

p > 0

.
(a) Show that the equation of the tangent at

P

can be written as

4y = 3px - 2p^3

.

(4 marks)

The tangent to this curve at

P

crosses the

x

-axis at the point

A

and the normal to this curve at

P

crosses the

x

-axis at the point

B

.
(b) Show that the ratio

PA : PB

4 : 3p

.

(8 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

The line

l_1

passes through the points

P(-3,\ 8)

and

Q(7,\ 3)

, as shown in Figure 1.
(a) Find the equation of

l_1

, giving your answer in the form

y = mx + c

, where

m

and

c

are constants to be found.

(3 marks)

The line

l_2

is perpendicular to

l_1

and passes through the origin

O

.

The point

R

is the point of intersection of

l_1

and

l_2

, as shown in Figure 1.
(b) Show that

R

has coordinates

\left(\tfrac{13}{5},\ \tfrac{26}{5}\right)

(4 marks)
(c) Find the exact area of triangle

OPQ

.

(3 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Given that

the point $A$ has coordinates $(2,\ -1)$
the point $B$ has coordinates $(8,\ 7)$
the line $l_1$ passes through $A$ and $B$

(a) find an equation for

l_1

, giving your answer in the form

px + qy + r = 0

where

p

q

and

r

are integers to be found.

(3 marks)

The line

l_2

passes through

B

and is parallel to the

x

-axis.

The point

C

lies on

l_2

so that the length of

AC

\sqrt{89}

(b) Show that the

x

coordinate of

C

satisfies

x^2 - 4x - 21 = 0

(3 marks)
(c) Hence find both possible pairs of coordinates of the point

C

.

(2 marks)
(d) Hence find the minimum possible area of triangle

ABC

.

(2 marks)

●●●●●Level 510 marksStart

The lines

L_1

and

L_2

are parallel.

L_1

has equation

3x + 2y = 8

and

L_2

has equation

3x + 2y = 34

L_1

intersects the

y

-axis at the point

P

The point

Q

is the point on

L_2

closest to

P

, as shown in the diagram.(a) (i) Find the coordinates of

Q

(ii) Hence, show that

PQ = k\sqrt{13}

, where

k

is an integer to be found.

(7 marks)
(b) A circle,

C

, has centre

(a, 3)

L_1

and

L_2

are both tangents to

C

(b) (i) Find

a

(ii) Find the equation of

C

(4 marks)

●●●●●Level 511 marksStart

The circle

C_1

has equation

x^2 + y^2 - 8x + 6y - 11 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(-2,\ 5)$
has radius $k$ , where $k$ is a positive constant

(b) Show that the distance between the centre of

C_1

and the centre of

C_2

10

.

(2 marks)
(c) Find the range of values of

k

for which

C_1

lies entirely inside

C_2

, writing your answer in set notation.

(2 marks)

Given instead that

C_1

and

C_2

intersect at 2 distinct points,
(d) find the range of values of

k

, writing your answer in set notation.

(4 marks)

●●●●●Level 511 marksStart

The equation of a circle

37 questions

LessonNot started

A circle has centre

C(1,3)

and passes through the point

P(4,7)

.
(a) Find the equation of the circle. (2 marks)
(b) Find the equation of the tangent to the circle at

P

, giving your answer in the form

ax + by = c

, where

a

b

and

c

are integers. (3 marks)

●●●●●Level 25 marksStart

A circle

C

has equation

x^2 + y^2 + 10kx - 4ky + 9 = 0

where

k

is a constant.

Find, in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

●●●●●Level 23 marksStart

The points

P

and

Q

have coordinates

(-1, 7)

and

(5, -3)

respectively.

Determine the equation of the circle that has

PQ

as a diameter. Give your answer in the form

x^2 + y^2 + ax + by + c = 0

, where

a

b

and

c

are integers.

(4 marks)

●●●●●Level 34 marksStart

For

0 \le t < 2\pi

, a curve

C

has parametric equations

x = 5 - k + 3k\sin t, \qquad y = -2 + 6\cos t,

where

k

is a non-zero constant.
(a) Find a cartesian equation for

C

. You do not need to simplify your answer.

(2 marks)

You are given that

C

is a circle.
(b) (i) Determine the radius of

C

.

(ii) Find the possible coordinates of the centre of

C

.

(4 marks)

●●●●●Level 36 marksStart

The circle

C_1

has equation

x^{2} + y^{2} - 6x + 10y = 11

(a) (i) Find the coordinates of the centre of

C_1

(ii) Find the exact value of the radius of

C_1

(3 marks)

In part (b) you must show detailed reasoning.

The circle

C_2

has equation

(x + 7)^{2} + (y - 4)^{2} = 20

(b) Prove that the circles

C_1

and

C_2

neither touch nor intersect.

(3 marks)

●●●●●Level 36 marksStart

The line with equation

y = 3x + c

, where

c

is a constant, is a tangent to the circle with equation

x^2 + y^2 = 10

Find the two possible values of

c

. (5 marks)

●●●●●Level 35 marksStart

The diagram below is a sketch and is not accurately drawn.The circle

C

touches the

y

-axis at the point

T

and passes through the point

A(2,\ 7)

, as shown in Figure 1.

The centre

K

C

lies on the line

l

with equation

y = 2x - 7

, and the

x

-coordinate of

K

p

, where

p > 0

.
(a) Explain why the radius of

C

p

. (1 mark)
(b) Show that

p^2 - 15p + 50 = 0

. (3 marks)

Given also that

C

crosses the

x

-axis,
(c) find an equation for

C

, justifying your choice of

p

. (3 marks)
(d) Find the length of the chord that

C

cuts off on the

x

-axis. (2 marks)

●●●●●Level 39 marksStart

A circle

C

has equation

x^2 + y^2 + 12x - 4y = 41

(a) Find

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The point

P

lies on

C

and is the point on

C

furthest away from the origin

O

.
(b) Find the exact length

OP

.

(2 marks)

●●●●●Level 35 marksStart

A circle has equation

x^2 + y^2 - 14x + 6y + 22 = 0

(a) Find

(i) the coordinates of the centre of the circle,

(ii) the radius of the circle.

(3 marks)

Given that

Q

is the point on the circle that is closest to the origin

O

,
(b) find the exact length

OQ

.

(2 marks)

●●●●●Level 35 marksStart

A circle has equation

x^2 + y^2 - 10x + 4y + 9 = 0

.
(a) Find the centre and radius of the circle.

(3 marks)
(b) Find the coordinates of any points where the line

x + 2y = 11

meets the circle.

(4 marks)
(c) State what can be deduced from the answer to part (b) about the line and the circle.

(1 mark)

●●●●●Level 48 marksStart

A circle with centre

C

has equation

x^2 + y^2 - 8x + 2y + 13 = 0

.
(a) Find

(i) the coordinates of

C

,

(ii) the radius of the circle.

(3 marks)
(b) Determine the set of values of

k

for which the line

y = kx + 1

does not intersect or touch the circle.

(5 marks)

●●●●●Level 48 marksStart

A circle

C

has equation

x^2 + y^2 - 4x + 6y + k = 0

.
(a) Find the set of possible values of

k

.

(2 marks)

It is given that

k = -7

.
(b) Determine the coordinates of the two points on

C

at which the gradient of the tangent is

2

.

(5 marks)

●●●●●Level 47 marksStart

The points

A

B

and

C

have coordinates

(-2, 6)

(3, 7)

and

(10, 0)

respectively.
(a) Find the equation of the perpendicular bisector of

AB

. Give your answer in the form

y = mx + c

.

(3 marks)
(b) In this question you must show detailed reasoning.

Hence, or otherwise, find an equation of the circle that passes through points

A

B

and

C

.

(5 marks)

●●●●●Level 48 marksStart

The points

A

and

B

have position vectors

\mathbf{a}

and

\mathbf{b}

respectively, relative to the origin

O

.
(a) State, with reference to the points

A

and

B

, the geometrical significance of

(i)

|\mathbf{a} - \mathbf{b}|

,

(ii)

\tfrac{1}{3}(\mathbf{a} + 2\mathbf{b})

.

(2 marks)

The point

P

has position vector

\mathbf{p}

, where

\left|\mathbf{p} - \tfrac{1}{3}(\mathbf{a} + 2\mathbf{b})\right| = \tfrac{1}{3}|\mathbf{a} - \mathbf{b}|

.
(b) (i) Describe the locus of

P

.

(ii) Explain why

B

lies on this locus.

(2 marks)

It is now given that

\mathbf{a} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}

and

\mathbf{b} = \begin{pmatrix} 7 \\ 7 \end{pmatrix}

.
(c) Find a cartesian equation of the locus of

P

.

(4 marks)

●●●●●Level 48 marksStart

The circle

C

has equation

x^{2} + y^{2} - 4x + 6y = 27

(a) Find

(i) the coordinates of the centre of

C

,

(ii) the exact radius of

C

.

(3 marks)

The point

P

lies on

C

.

Given that the tangent to

C

P

has equation

x + 3y - 13 = 0

(b) find the coordinates of

P

(4 marks)
(c) Find the equation of the normal to

C

P

, giving your answer in the form

y = mx + c

where

m

and

c

are integers to be found.

(3 marks)

●●●●●Level 410 marksStart

The circle

C

has equation

x^{2} + y^{2} - 10x + 6y + 21 = 0

(a) Find

(i) the coordinates of the centre of

C

,

(ii) the exact value of the radius of

C

.

(3 marks)

The line

L

has equation

y = mx - 4

, where

m

is a constant.

Given that

L

is the tangent to

C

at the point

P

,
(b) show that

6m^{2} - 5m - 6 = 0

(3 marks)
(c) Hence find the possible pairs of coordinates of

P

.

(4 marks)

●●●●●Level 410 marksStart

The line joining the points

(-1, -4)

and

(5, -10)

is the diameter of a circle

C

.
(a) Find an equation for

C

.

(5 marks)
(b) Hence find the exact coordinates of the point on

C

that is nearest the

x

-axis.

(2 marks)

●●●●●Level 47 marksStart

The circle

C

has centre

(2, -3)

and radius

5

.

The line

l

has equation

y = 2x + k

, where

k

is a constant.

Find the exact values of

k

for which

l

is a tangent to

C

. (6 marks)

●●●●●Level 46 marksStart

Figure 1 shows the circle

C

with equation

x^2 + y^2 - 8x + 2y - 8 = 0

together with the tangent to

C

at the point

P(7, 3)

.Figure 1 (not accurately drawn)

Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers. (5 marks)

●●●●●Level 45 marksStart

The circle

C

has equation

x^2 + y^2 - 10x - 6y + k = 0

where

k

is a constant.

The line

l

with equation

y = 3x - 2

is a tangent to

C

.
(a) Find the value of

k

. (3 marks)
(b) Find the coordinates of the point at which

l

touches

C

. (2 marks)

●●●●●Level 45 marksStart

The circle

C

has centre

(5, -3)

and passes through the point

A(2, 1)

.

The line with equation

y = x + k

, where

k

is a constant, is a tangent to

C

.

Find the possible values of

k

, giving your answers in exact form. (5 marks)

●●●●●Level 45 marksStart

The points

P

and

Q

have coordinates

(2, 9)

and

(8, -3)

respectively.
(a) (i) Find the coordinates of the midpoint of

PQ

(ii) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(5 marks)
(b) A circle passes through the points

P

and

Q

The centre of the circle lies on the line with equation

3x - y = 17

(b) (i) Find the equation of the circle.

(ii) The circle intersects the coordinate axes at

n

points.

State the value of

n

(4 marks)

●●●●●Level 49 marksStart

Figure 1 below shows a circle.

The centre of the circle is

P

and the circle intersects the

y

-axis at

Q

as shown in Figure 1.

Figure 1The equation of the circle is

x^2 + y^2 = 6y - 10x + 7

(a) Express the equation of the circle in the form

(x - a)^2 + (y - b)^2 = k

where

a

b

and

k

are constants to be found.

(3 marks)
(b) State the coordinates of

P

(1 mark)
(c) Find the

y

-coordinate of

Q

(2 marks)
(d) The line segment

QR

is a tangent to the circle as shown in Figure 2 below.

The point

R

has coordinates

(12, -8)

Figure 2Find the angle

QPR

Give your answer in radians to three significant figures.

(3 marks)

●●●●●Level 49 marksStart

Figure 1 below shows a circle.

The centre of the circle is

P

and the circle intersects the

y

-axis at

Q

as shown in Figure 1.

Figure 1The equation of the circle is

x^2 + y^2 = 10x - 8y - 7

(a) Express the equation of the circle in the form

(x - a)^2 + (y - b)^2 = k

where

a

b

and

k

are constants to be found.

(3 marks)
(b) State the coordinates of

P

(1 mark)
(c) Find the

y

-coordinate of

Q

(2 marks)
(d) The line segment

QR

is a tangent to the circle as shown in Figure 2 below.

The point

R

has coordinates

(3, 4)

Figure 2Find the angle

QPR

Give your answer in radians to three significant figures.

(3 marks)

●●●●●Level 49 marksStart

A circle

C

has equation

x^2 + y^2 + 4kx - 8ky + 20 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line with equation

y = x - 2

intersects

C

at 2 distinct points.
(b) Find the range of possible values of

k

.

(6 marks)

●●●●●Level 49 marksStart

A circle

C

has equation

x^2 + y^2 + 8kx - 6ky + 5 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line with equation

y = 2x + 1

intersects

C

at 2 distinct points.
(b) Find the range of possible values of

k

.

(6 marks)

●●●●●Level 49 marksStart

The circle

C_1

has equation

x^2 + y^2 + 10x - 6y + 9 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(7,\ 8)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 - 12x + 2y + 28 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates

(-2,\ 5)

and radius

k

, where

k

is a positive constant.

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 + 2x - 14y + 34 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(8,\ 19)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

have no points in common,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

In this question you must show detailed reasoning.

The diagram shows the circle with equation

x^2 + y^2 - 12x - 8y + 42 = 0

and the point

P(0, -4)

. The tangent from

P

touching the circle at

A

has equation

y = mx - 4

, where

m > 1

.(a) (i) Show that the

x

-coordinate of

A

satisfies

(m^2 + 1)x^2 - 4(4m + 3)x + 90 = 0

.

(ii) Hence determine the equation of this tangent.

(6 marks)

The other tangent from

P

touches the circle at

B

and has equation

y = nx - 4

, where

n < 1

.
(b) Determine the exact value of

\tan APB

.

(4 marks)

●●●●●Level 510 marksStart

The diagram shows the circle with equation

x^2 + y^2 - 8y + a = 0

, where

a

is a constant. The centre

C

of the circle lies on the

y

-axis. The line

y = 2x

meets the circle at the points

A

and

B

, and

M

is the midpoint of

AB

.(a) In this question you must show detailed reasoning.

Show that the area of triangle

ABC

\tfrac45\sqrt{64 - 5a}

.

(7 marks)
(b) (i) Find the value of

a

for which this area is zero.

(ii) Give a geometrical interpretation of the case in part (b)(i).

(2 marks)
(c) Give a geometrical interpretation of the case

a = 13

.

(1 mark)

●●●●●Level 510 marksStart

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.Figure 1

Figure 1 shows

the curve $C_1$ with equation $y = x^{3} - 3x + 4$
the circle $C_2$ with centre $T$

The point

T

is the minimum turning point of

C_1

Using Figure 1 and calculus,
(a) find the coordinates of

T

(3 marks)

The point

A

, with

x

coordinate

2

, lies on both

C_1

and

C_2

(b) Find an equation of the circle

C_2

(3 marks)

The line

l

, shown in Figure 1, is the tangent to circle

C_2

A

.
(c) Show that an equation of

l

x + 4y = 26

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C_1

l

and the

y

-axis.
(d) Find the exact area of

R

.

(3 marks)

●●●●●Level 512 marksStart

A circle passes through

A(1, 1)

B(1, 7)

and

C(7, 7)

. Find its equation.

●●●●●Level 54 marksStart

The equation

x^2 + y^2 - 8x + ky + 9 = 0

represents a circle of radius

4

. Find the possible values of

k

●●●●●Level 53 marksStart

The lines

L_1

and

L_2

are parallel.

L_1

has equation

3x + 2y = 8

and

L_2

has equation

3x + 2y = 34

L_1

intersects the

y

-axis at the point

P

The point

Q

is the point on

L_2

closest to

P

, as shown in the diagram.(a) (i) Find the coordinates of

Q

(ii) Hence, show that

PQ = k\sqrt{13}

, where

k

is an integer to be found.

(7 marks)
(b) A circle,

C

, has centre

(a, 3)

L_1

and

L_2

are both tangents to

C

(b) (i) Find

a

(ii) Find the equation of

C

(4 marks)

●●●●●Level 511 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A circle

C

has equation

x^2 + y^2 + 4kx - 2ky + 15 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line

l

has equation

y = 3x + 2

(b) Show that the

x

coordinates of any points of intersection of

l

and

C

satisfy

10x^2 + (12 - 2k)x + 19 - 4k = 0

(3 marks)

Given that

l

intersects

C

at 2 distinct points,
(c) find the range of possible values of

k

.

(4 marks)

Given also that

k

is a negative integer,
(d) state the greatest possible value of

k

.

(1 mark)

●●●●●Level 511 marksStart

The circle

C_1

has equation

x^2 + y^2 - 8x + 6y - 11 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(-2,\ 5)$
has radius $k$ , where $k$ is a positive constant

(b) Show that the distance between the centre of

C_1

and the centre of

C_2

10

.

(2 marks)
(c) Find the range of values of

k

for which

C_1

lies entirely inside

C_2

, writing your answer in set notation.

(2 marks)

Given instead that

C_1

and

C_2

intersect at 2 distinct points,
(d) find the range of values of

k

, writing your answer in set notation.

(4 marks)

●●●●●Level 511 marksStart

Lines and circles — tangents, chords, and intersections

35 questions

LessonNot started

A circle has centre

C(1,3)

and passes through the point

P(4,7)

.
(a) Find the equation of the circle. (2 marks)
(b) Find the equation of the tangent to the circle at

P

, giving your answer in the form

ax + by = c

, where

a

b

and

c

are integers. (3 marks)

●●●●●Level 25 marksStart

The circle

C_1

has equation

x^{2} + y^{2} - 6x + 10y = 11

(a) (i) Find the coordinates of the centre of

C_1

(ii) Find the exact value of the radius of

C_1

(3 marks)

In part (b) you must show detailed reasoning.

The circle

C_2

has equation

(x + 7)^{2} + (y - 4)^{2} = 20

(b) Prove that the circles

C_1

and

C_2

neither touch nor intersect.

(3 marks)

●●●●●Level 36 marksStart

The circle

C

has equation

x^2 + y^2 - 4x - 6y + 8 = 0

.
(a) Find its centre and radius.
(b) Find the equation of the tangent to

C

at the point

P(3, 5)

, in the form

ax + by + c = 0

●●●●●Level 35 marksStart

The diagram below is a sketch and is not accurately drawn.The line

y = mx

passes through the origin and is a tangent to the circle

(x - 6)^2 + (y - 3)^2 = 9

. Find the possible values of

m

●●●●●Level 34 marksStart

Find the length of the tangent from the point

P(10, 4)

to the circle

(x - 2)^2 + (y - 1)^2 = 9

●●●●●Level 33 marksStart

The diagram below is a sketch and is not accurately drawn.The circle

C

touches the

y

-axis at the point

T

and passes through the point

A(2,\ 7)

, as shown in Figure 1.

The centre

K

C

lies on the line

l

with equation

y = 2x - 7

, and the

x

-coordinate of

K

p

, where

p > 0

.
(a) Explain why the radius of

C

p

. (1 mark)
(b) Show that

p^2 - 15p + 50 = 0

. (3 marks)

Given also that

C

crosses the

x

-axis,
(c) find an equation for

C

, justifying your choice of

p

. (3 marks)
(d) Find the length of the chord that

C

cuts off on the

x

-axis. (2 marks)

●●●●●Level 39 marksStart

The diagram below is a sketch and is not accurately drawn.Figure 4 shows the circle

C

with equation

(x-3)^2 + (y-2)^2 = 5

and a line

l

with equation

y = 2x + c

.

Given that

l

is a tangent to

C

, find the possible values of

c

●●●●●Level 34 marksStart

The circle

C

has equation

(x - 3)^2 + (y - 2)^2 = 25

. The point

P(6, 6)

lies on

C

.(a) Verify that

P

lies on

C

. (1 mark)
(b) Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by = c

. (5 marks)

●●●●●Level 36 marksStart

(a) Show that the line

y = 2x - 5

is a tangent to the circle

x^2 + y^2 = 5

. (4 marks)
(b) Find the coordinates of the point of contact. (2 marks)

●●●●●Level 36 marksStart

A circle has centre

(2, 3)

and passes through the point

(5, 7)

.
(a) Find the radius of the circle. (2 marks)
(b) Write down an equation of the circle. (2 marks)

●●●●●Level 34 marksStart

The points

A(1, 2)

and

B(7, 10)

are the ends of a diameter of a circle.
(a) Find the centre and radius. (3 marks)
(b) Find an equation of the circle. (2 marks)
(c) The point

D(7, 2)

lies on the circle. State, with a reason, the size of angle

ADB

. (2 marks)

●●●●●Level 37 marksStart

A circle has equation

x^2 + y^2 - 10x + 4y + 9 = 0

.
(a) Find the centre and radius of the circle.

(3 marks)
(b) Find the coordinates of any points where the line

x + 2y = 11

meets the circle.

(4 marks)
(c) State what can be deduced from the answer to part (b) about the line and the circle.

(1 mark)

●●●●●Level 48 marksStart

A circle with centre

C

has equation

x^2 + y^2 - 8x + 2y + 13 = 0

.
(a) Find

(i) the coordinates of

C

,

(ii) the radius of the circle.

(3 marks)
(b) Determine the set of values of

k

for which the line

y = kx + 1

does not intersect or touch the circle.

(5 marks)

●●●●●Level 48 marksStart

A circle

C

has equation

x^2 + y^2 - 4x + 6y + k = 0

.
(a) Find the set of possible values of

k

.

(2 marks)

It is given that

k = -7

.
(b) Determine the coordinates of the two points on

C

at which the gradient of the tangent is

2

.

(5 marks)

●●●●●Level 47 marksStart

The points

A

B

and

C

have coordinates

(-2, 6)

(3, 7)

and

(10, 0)

respectively.
(a) Find the equation of the perpendicular bisector of

AB

. Give your answer in the form

y = mx + c

.

(3 marks)
(b) In this question you must show detailed reasoning.

Hence, or otherwise, find an equation of the circle that passes through points

A

B

and

C

.

(5 marks)

●●●●●Level 48 marksStart

The circle

C

has equation

x^{2} + y^{2} - 4x + 6y = 27

(a) Find

(i) the coordinates of the centre of

C

,

(ii) the exact radius of

C

.

(3 marks)

The point

P

lies on

C

.

Given that the tangent to

C

P

has equation

x + 3y - 13 = 0

(b) find the coordinates of

P

(4 marks)
(c) Find the equation of the normal to

C

P

, giving your answer in the form

y = mx + c

where

m

and

c

are integers to be found.

(3 marks)

●●●●●Level 410 marksStart

The circle

C

has equation

x^{2} + y^{2} - 10x + 6y + 21 = 0

(a) Find

(i) the coordinates of the centre of

C

,

(ii) the exact value of the radius of

C

.

(3 marks)

The line

L

has equation

y = mx - 4

, where

m

is a constant.

Given that

L

is the tangent to

C

at the point

P

,
(b) show that

6m^{2} - 5m - 6 = 0

(3 marks)
(c) Hence find the possible pairs of coordinates of

P

.

(4 marks)

●●●●●Level 410 marksStart

Figure 1 shows the circle

C

with equation

x^2 + y^2 - 8x + 2y - 8 = 0

together with the tangent to

C

at the point

P(7, 3)

.Figure 1 (not accurately drawn)

Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers. (5 marks)

●●●●●Level 45 marksStart

The circle

C

has equation

x^2 + y^2 - 10x - 6y + k = 0

where

k

is a constant.

The line

l

with equation

y = 3x - 2

is a tangent to

C

.
(a) Find the value of

k

. (3 marks)
(b) Find the coordinates of the point at which

l

touches

C

. (2 marks)

●●●●●Level 45 marksStart

The circle

C

has centre

(5, -3)

and passes through the point

A(2, 1)

.

The line with equation

y = x + k

, where

k

is a constant, is a tangent to

C

.

Find the possible values of

k

, giving your answers in exact form. (5 marks)

●●●●●Level 45 marksStart

The points

P

and

Q

have coordinates

(2, 9)

and

(8, -3)

respectively.
(a) (i) Find the coordinates of the midpoint of

PQ

(ii) Find the equation of the perpendicular bisector of

PQ

Give your answer in the form

ax + by = c

where

a

b

and

c

are integers.

(5 marks)
(b) A circle passes through the points

P

and

Q

The centre of the circle lies on the line with equation

3x - y = 17

(b) (i) Find the equation of the circle.

(ii) The circle intersects the coordinate axes at

n

points.

State the value of

n

(4 marks)

●●●●●Level 49 marksStart

Figure 1 below shows a circle.

The centre of the circle is

P

and the circle intersects the

y

-axis at

Q

as shown in Figure 1.

Figure 1The equation of the circle is

x^2 + y^2 = 6y - 10x + 7

(a) Express the equation of the circle in the form

(x - a)^2 + (y - b)^2 = k

where

a

b

and

k

are constants to be found.

(3 marks)
(b) State the coordinates of

P

(1 mark)
(c) Find the

y

-coordinate of

Q

(2 marks)
(d) The line segment

QR

is a tangent to the circle as shown in Figure 2 below.

The point

R

has coordinates

(12, -8)

Figure 2Find the angle

QPR

Give your answer in radians to three significant figures.

(3 marks)

●●●●●Level 49 marksStart

Figure 1 below shows a circle.

The centre of the circle is

P

and the circle intersects the

y

-axis at

Q

as shown in Figure 1.

Figure 1The equation of the circle is

x^2 + y^2 = 10x - 8y - 7

(a) Express the equation of the circle in the form

(x - a)^2 + (y - b)^2 = k

where

a

b

and

k

are constants to be found.

(3 marks)
(b) State the coordinates of

P

(1 mark)
(c) Find the

y

-coordinate of

Q

(2 marks)
(d) The line segment

QR

is a tangent to the circle as shown in Figure 2 below.

The point

R

has coordinates

(3, 4)

Figure 2Find the angle

QPR

Give your answer in radians to three significant figures.

(3 marks)

●●●●●Level 49 marksStart

A circle

C

has equation

x^2 + y^2 + 4kx - 8ky + 20 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line with equation

y = x - 2

intersects

C

at 2 distinct points.
(b) Find the range of possible values of

k

.

(6 marks)

●●●●●Level 49 marksStart

A circle

C

has equation

x^2 + y^2 + 8kx - 6ky + 5 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line with equation

y = 2x + 1

intersects

C

at 2 distinct points.
(b) Find the range of possible values of

k

.

(6 marks)

●●●●●Level 49 marksStart

The circle

C_1

has equation

x^2 + y^2 + 10x - 6y + 9 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(7,\ 8)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 - 12x + 2y + 28 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates

(-2,\ 5)

and radius

k

, where

k

is a positive constant.

Given that

C_1

and

C_2

intersect at 2 distinct points,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

The circle

C_1

has equation

x^2 + y^2 + 2x - 14y + 34 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(8,\ 19)$
has radius $k$ , where $k$ is a positive constant

Given that

C_1

and

C_2

have no points in common,
(b) find the range of values of

k

, writing your answer in set notation.

(5 marks)

●●●●●Level 48 marksStart

In this question you must show detailed reasoning.

The diagram shows the circle with equation

x^2 + y^2 - 12x - 8y + 42 = 0

and the point

P(0, -4)

. The tangent from

P

touching the circle at

A

has equation

y = mx - 4

, where

m > 1

.(a) (i) Show that the

x

-coordinate of

A

satisfies

(m^2 + 1)x^2 - 4(4m + 3)x + 90 = 0

.

(ii) Hence determine the equation of this tangent.

(6 marks)

The other tangent from

P

touches the circle at

B

and has equation

y = nx - 4

, where

n < 1

.
(b) Determine the exact value of

\tan APB

.

(4 marks)

●●●●●Level 510 marksStart

The diagram shows the circle with equation

x^2 + y^2 - 8y + a = 0

, where

a

is a constant. The centre

C

of the circle lies on the

y

-axis. The line

y = 2x

meets the circle at the points

A

and

B

, and

M

is the midpoint of

AB

.(a) In this question you must show detailed reasoning.

Show that the area of triangle

ABC

\tfrac45\sqrt{64 - 5a}

.

(7 marks)
(b) (i) Find the value of

a

for which this area is zero.

(ii) Give a geometrical interpretation of the case in part (b)(i).

(2 marks)
(c) Give a geometrical interpretation of the case

a = 13

.

(1 mark)

●●●●●Level 510 marksStart

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.Figure 1

Figure 1 shows

the curve $C_1$ with equation $y = x^{3} - 3x + 4$
the circle $C_2$ with centre $T$

The point

T

is the minimum turning point of

C_1

Using Figure 1 and calculus,
(a) find the coordinates of

T

(3 marks)

The point

A

, with

x

coordinate

2

, lies on both

C_1

and

C_2

(b) Find an equation of the circle

C_2

(3 marks)

The line

l

, shown in Figure 1, is the tangent to circle

C_2

A

.
(c) Show that an equation of

l

x + 4y = 26

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C_1

l

and the

y

-axis.
(d) Find the exact area of

R

.

(3 marks)

●●●●●Level 512 marksStart

A circle has equation

x^2 + y^2 - 2x + 8y - 8 = 0

.
(a) Find the coordinates of the centre and the radius. (4 marks)
(b) Determine whether the point

(5, -1)

lies inside, on, or outside the circle. (3 marks)

●●●●●Level 57 marksStart

The lines

L_1

and

L_2

are parallel.

L_1

has equation

3x + 2y = 8

and

L_2

has equation

3x + 2y = 34

L_1

intersects the

y

-axis at the point

P

The point

Q

is the point on

L_2

closest to

P

, as shown in the diagram.(a) (i) Find the coordinates of

Q

(ii) Hence, show that

PQ = k\sqrt{13}

, where

k

is an integer to be found.

(7 marks)
(b) A circle,

C

, has centre

(a, 3)

L_1

and

L_2

are both tangents to

C

(b) (i) Find

a

(ii) Find the equation of

C

(4 marks)

●●●●●Level 511 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A circle

C

has equation

x^2 + y^2 + 4kx - 2ky + 15 = 0

where

k

is a constant.
(a) Find in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

(3 marks)

The line

l

has equation

y = 3x + 2

(b) Show that the

x

coordinates of any points of intersection of

l

and

C

satisfy

10x^2 + (12 - 2k)x + 19 - 4k = 0

(3 marks)

Given that

l

intersects

C

at 2 distinct points,
(c) find the range of possible values of

k

.

(4 marks)

Given also that

k

is a negative integer,
(d) state the greatest possible value of

k

.

(1 mark)

●●●●●Level 511 marksStart

The circle

C_1

has equation

x^2 + y^2 - 8x + 6y - 11 = 0

(a) Find

(i) the coordinates of the centre of

C_1

(ii) the radius of

C_1

(3 marks)

A different circle

C_2

has centre with coordinates $(-2,\ 5)$
has radius $k$ , where $k$ is a positive constant

(b) Show that the distance between the centre of

C_1

and the centre of

C_2

10

.

(2 marks)
(c) Find the range of values of

k

for which

C_1

lies entirely inside

C_2

, writing your answer in set notation.

(2 marks)

Given instead that

C_1

and

C_2

intersect at 2 distinct points,
(d) find the range of values of

k

, writing your answer in set notation.

(4 marks)

●●●●●Level 511 marksStart

Parametric curves

19 questions

LessonNot started

A curve has parametric equations

x = 3t,\ y = t^2 + 1

.
(a) Find the coordinates of the points where

t = 2

and

t = -1

.
(b) The curve passes through the point

(12, k)

. Find the value of

t

there and the value of

k

●●●●●Level 24 marksStart

A curve has parametric equations

x = t - 2,\ y = t^2 + t

. Find the coordinates of the points on the curve where

t = -2,\ 0,\ 1

and

3

●●●●●Level 22 marksStart

A curve has parametric equations

x = 2t - 1, \; y = t^{2} + 3

.
(a) Find the coordinates of the point where

t = 2

. (1 mark)
(b) Find the value of

y

at the point where

x = 5

. (2 marks)

●●●●●Level 23 marksStart

A curve has parametric equations

x = 2t,\ y = t^2 - 4t

for

-1 \le t \le 5

. Find the range of values of

x

and the range of values of

y

●●●●●Level 33 marksStart

A curve has parametric equations

x = t + 1,\ y = \dfrac{4}{t}

t \ne 0

.
(a) Find the coordinates of the point where

t = 2

.
(b) Find the value of

t

for which

x = 5

, and the corresponding

y

.
(c) Explain why the curve never meets the

x

-axis.

●●●●●Level 34 marksStart

A curve

C

has parametric equations

x = t^2,\ y = t^3

t \in \mathbb{R}

.
(a) Show that both

(4, 8)

and

(4, -8)

lie on

C

, stating the value of

t

at each.
(b) Explain why no point of

C

has a negative

x

-coordinate.

●●●●●Level 33 marksStart

A curve

C

has parametric equations

x = t^2 - 3,\ y = t^3 - 3t

.
(a) Find the coordinates of the points on

C

where

t = -2, -1, 0, 1

and

2

.
(b) Show that

C

is symmetric about the

x

-axis.

●●●●●Level 34 marksStart

A curve has parametric equations

x = t - \dfrac{3}{t},\ y = t + \dfrac{3}{t}

, for

t > 0

. Find the value of

t

for which

x = 2

, and the coordinates of the corresponding point.

●●●●●Level 33 marksStart

A curve has parametric equations

x = t^2 + 1,\ y = 2t

t \in \mathbb{R}

.
(a) Find the coordinates of the point where

t = 0

.
(b) Find the least possible value of

x

and the value of

y

there.
(c) Hence describe the shape of the curve.

●●●●●Level 34 marksStart

A curve has parametric equations

x = t^{2}, \; y = 4t

.
(a) Show that the point

(9, 12)

lies on the curve. (2 marks)
(b) Determine whether the point

(4, 10)

lies on the curve. (2 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = 3 + 2\sin t, \qquad y = 2\cos t, \qquad 0 \leqslant t < 2\pi

State the range of possible values of

x

and of

y

. (3 marks)

●●●●●Level 33 marksStart

A curve has parametric equations

x = t^{2} - 4, \; y = t - 1

.
(a) Find the coordinates of the points where the curve crosses the

y

-axis. (3 marks)
(b) Find the coordinates of the point where the curve crosses the

x

-axis. (2 marks)

●●●●●Level 35 marksStart

A curve has parametric equations

x = t^{2} - 6t, \; y = 2t

.

Find the least possible value of

x

, and the coordinates of the point on the curve where it occurs. (4 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = 2t + 1, \; y = t^{2} - 1

for

0 \leqslant t \leqslant 3

.
(a) Find the coordinates of the two endpoints of the curve. (2 marks)
(b) Find the coordinates of the point where the curve meets the

x

-axis. (2 marks)
(c) State the range of values of

x

on the curve. (1 mark)

●●●●●Level 35 marksStart

The points

P

and

Q

on the curve

x = t^{2}, \; y = 2t

correspond to

t = 1

and

t = 3

respectively.

Find the exact distance

PQ

, giving your answer in the form

k\sqrt{5}

. (4 marks)

●●●●●Level 34 marksStart

A curve

C

has parametric equations

x = t^2 + 1, \qquad y = t^3 - 4t, \qquad t \in \mathbb{R}

(a) Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

. (3 marks)
(b) Find the exact coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (4 marks)
(c) Show that

C

crosses itself, and find the coordinates of the point at which it does so. (2 marks)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of part of the curve defined by the parametric equations

x = t^2 - 1 \qquad y = \dfrac{5}{(t + 1)(t + 4)} \qquad t \geqslant 0

The region

R

, shown shaded in Figure 1, is bounded by the curve, the

y

-axis, the

x

-axis and the line with equation

x = 8

The curve meets the

y

-axis where

t = a

and meets the line with equation

x = 8

where

t = b

.
(a) Find the value of

a

and the value of

b

.

(2 marks)
(b) Show that the area of region

R

is given by

M\int_a^b \dfrac{t}{(t + 1)(t + 4)}\,dt

where

M

is a constant to be found.

(5 marks)
(c) (i) Write

\dfrac{t}{(t + 1)(t + 4)}

in partial fractions.

(ii) Use algebraic integration to find the exact area of

R

, giving your answer in the form

p\ln\dfrac{7}{5} + q\ln 2

where

p

and

q

are rational constants.

(6 marks)

●●●●●Level 513 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with parametric equations

x = t^2 + 2t \qquad y = 2t - \dfrac{2}{t} \qquad t > 0

The curve

C

intersects the

x

-axis at the point

Q

.
(a) Find the

x

coordinate of

Q

.

(1 mark)

The line

l

is the normal to

C

at the point

P

as shown in Figure 1.

Given that

t = 2

P

(b) write down the coordinates of

P

(1 mark)
(c) Using calculus, show that an equation of

l

12x + 5y = 111

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by the curve

C

, the line

l

and the

x

-axis.
(d) Using algebraic integration, find the exact volume of the solid of revolution formed when the region

R

is rotated through

2\pi

radians about the

x

-axis. Give your answer in the form

\pi(p + q\ln 2)

where

p

and

q

are rational constants.

(7 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = t^2 + 1, \qquad y = t^3 - 4t,

where

t

takes all real values.
(a) The curve meets the

x

-axis at the points

A

and

B

, where

A

is closer to the origin. Find the coordinates of

A

and

B

, and show that the curve passes through

B

for two distinct values of

t

. (3 marks)
(b) The curve has a loop. Determine the exact area of the region enclosed by the loop. (5 marks)

●●●●●Level 58 marksStart

Converting to Cartesian form

27 questions

LessonNot started

A curve has parametric equations

x = t + 1,\ y = t^2

. Find a Cartesian equation of the curve.

●●●●●Level 22 marksStart

A curve has parametric equations

x = 2t,\ y = t^2 - 3

. Find a Cartesian equation of the curve.

●●●●●Level 22 marksStart

A curve has parametric equations

x = t - 3, \; y = 2t + 1

.

Find a Cartesian equation of the curve, and state what kind of curve it is. (3 marks)

●●●●●Level 23 marksStart

A curve has parametric equations

x = 3t, \; y = t^{2}

.

Find a Cartesian equation of the curve. (3 marks)

●●●●●Level 23 marksStart

The curve

C

has parametric equations

x = t^2 - 8t + 5 \qquad\qquad y = 10\ln(t - 4) \qquad\qquad t > 4

Find a Cartesian equation for

C

, giving your answer in the form

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

●●●●●Level 33 marksStart

For

0 \le t < 2\pi

, a curve

C

has parametric equations

x = 5 - k + 3k\sin t, \qquad y = -2 + 6\cos t,

where

k

is a non-zero constant.
(a) Find a cartesian equation for

C

. You do not need to simplify your answer.

(2 marks)

You are given that

C

is a circle.
(b) (i) Determine the radius of

C

.

(ii) Find the possible coordinates of the centre of

C

.

(4 marks)

●●●●●Level 36 marksStart

A curve

C

has parametric equations

x = 2\cos t

\quad y = \cos 2t

\quad 0 \le t \le \pi

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\cos t

. (3 marks)
(b) Find the Cartesian equation of

C

, stating the range of values of

x

for which the curve is defined. (3 marks)
(c) Verify that differentiating your Cartesian equation gives a result consistent with part (a). (1 mark)

●●●●●Level 37 marksStart

A curve has parametric equations

x = t + 1,\ y = \dfrac{1}{t}

t \ne 0

. Find a Cartesian equation of the curve and state any restriction on

x

●●●●●Level 33 marksStart

A curve has parametric equations

x = \sqrt{t},\ y = t - 4

for

t \ge 0

. Find the Cartesian equation of the curve and state its domain.

●●●●●Level 33 marksStart

A curve has parametric equations

x = \dfrac{2}{t},\ y = t + 1

t \ne 0

. Find a Cartesian equation of the curve and state any restriction on

x

●●●●●Level 33 marksStart

A curve has parametric equations

x = \sqrt{t - 1}, \; y = 3t

for

t \geqslant 1

.

Find a Cartesian equation of the curve, stating the range of valid

x

-values. (4 marks)

●●●●●Level 34 marksStart

The curve

C

has parametric equations

x = t^2 - 4t - 12 \qquad\qquad y = 8\ln(t - 2) \qquad\qquad t > 2

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the normal to

C

P

can be written in the form

ax + by = c\ln 2

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = t^2 - 14t + 40 \qquad\qquad y = 12\ln(t - 7) \qquad\qquad t > 7

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the tangent to

C

P

can be written in the form

ax + by = c\ln 3

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

A curve has parametric equations

x = \dfrac{2t^2 + 1}{t}

and

y = \dfrac{t^2 - 3}{t}

, for

t \ne 0

.
(a) Find

\dfrac{dy}{dx}

in terms of

t

, giving your answer in its simplest form.

(4 marks)
(b) Explain why the curve has no stationary points.

(2 marks)
(c) By considering

x - 2y

, or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.

(4 marks)

●●●●●Level 410 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = 5 - 2\cos 2\theta \qquad y = 6\sin\theta \qquad 0 \leqslant \theta \leqslant \dfrac{\pi}{2}

(a) Show that

\dfrac{dy}{dx} = k\operatorname{cosec}\theta \qquad \theta \neq 0

where

k

is a constant to be found.

(3 marks)

The point

P

lies on

C

where

\theta = \dfrac{\pi}{6}

(b) Find the equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

where

a

b

and

c

are integers.

(3 marks)
(c) Show that

C

has Cartesian equation

y^2 = 9(x - q) \qquad 3 \leqslant x \leqslant 7

where

q

is a constant to be found.

(3 marks)

●●●●●Level 49 marksStart

The curve

C

has parametric equations

x = \dfrac{2}{t+1}

\quad y = \dfrac{3t}{t+2}

\quad t > 0

.
(a) Show that a Cartesian equation of

C

y = \dfrac{6 - 3x}{x + 2}

. (3 marks)
(b) Find

\dfrac{dy}{dx}

in terms of

t

, and hence find the gradient of

C

at the point where

t = 1

. (3 marks)
(c) Find the range of values of

x

for points on

C

. (2 marks)

●●●●●Level 48 marksStart

The curve

C

has parametric equations

x = 4\sin t, \qquad y = 3\cos 2t - 1, \qquad -\dfrac{\pi}{2} \leqslant t \leqslant \dfrac{\pi}{2}

Figure 2 shows a sketch of

C

. The region

R

, shown shaded, is bounded by

C

and the

x

-axis.Figure 2
(a) Show that the Cartesian equation of

C

can be written as

y = a + bx^2

, where

a

and

b

are rational constants to be found, and state the domain of

x

. (4 marks)
(b) Find the exact area of

R

. (5 marks)

●●●●●Level 49 marksStart

A curve

C

has parametric equations

x = t^2, \qquad y = 2t, \qquad 0 \leqslant t \leqslant 3

(a) Show that the Cartesian equation of

C

y^2 = 4x

. (2 marks)
(b) Find the area of the region bounded by

C

, the

x

-axis and the line

x = 9

. (4 marks)
(c) Find an equation of the normal to

C

at the point where

t = 2

. (4 marks)

●●●●●Level 410 marksStart

A curve has parametric equations

x = t - 1,\ y = t^2 - 2t

. Show that the Cartesian equation of the curve is

y = x^2 - 1

●●●●●Level 42 marksStart

A curve has parametric equations

x = t^2,\ y = t^3

. Find a Cartesian equation of the curve in the form

y^2 = f(x)

, and state the restriction on

x

●●●●●Level 43 marksStart

A curve has parametric equations

x = \dfrac{1}{t + 1},\ y = \dfrac{t}{t + 1}

t \ne -1

. Show that the curve is part of a straight line, and state which single point of that line is excluded.

●●●●●Level 43 marksStart

A curve has parametric equations

x = \mathrm{e}^{t}, \; y = \mathrm{e}^{2t} + 1

.

Find a Cartesian equation of the curve, stating the range of valid

x

-values. (4 marks)

●●●●●Level 44 marksStart

A curve has parametric equations

x = \dfrac{1}{t - 2}, \; y = t + 3

\; t \neq 2

.

Find a Cartesian equation of the curve in the form

y = f(x)

. (4 marks)

●●●●●Level 44 marksStart

A curve has parametric equations

x = t + 2, \; y = t^{2} + 4t + 5

.

Show that the curve is the parabola

y = x^{2} + 1

. (4 marks)

●●●●●Level 44 marksStart

A furniture maker is designing a rocking chair, shown in Figure 1.

Figure 1The curve of each runner of the rocking chair can be modelled by the parametric equations

x = 20t \qquad y = 2t^2 + \frac{32}{t}

for

1 \leq t \leq 4

, as shown in Figure 2, where

x

and

y

are measured in centimetres.

P

is the point where

t = 1

and

Q

is the point where

t = 4

Figure 2(a) Show that a Cartesian equation of the curve is

y = \frac{x^2}{200} + \frac{640}{x}

(2 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) The chair rests on horizontal ground, with each runner touching the ground at its lowest point.

Hence, find the height of

Q

above the ground.

(6 marks)
(c) Suggest one limitation of the model.

(1 mark)

●●●●●Level 49 marksStart

The curve

C

has parametric equations

x = t + \dfrac{1}{t}

y = t - \dfrac{1}{t}

t > 0

(a) Show that a Cartesian equation of

C

x^2 - y^2 = 4

. (2 marks)
(b) Find an equation of the normal to

C

at the point where

t = 2

, giving your answer in the form

ax + by = c

, where

a

b

and

c

are integers. (4 marks)
(c) The finite region

R

is bounded by

C

, the

x

-axis and the normal found in part (b).

Find the exact area of

R

. (5 marks)

●●●●●Level 511 marksStart

A curve has parametric equations

x = t^{2} + 1, \; y = t^{4}

\; t \in \mathbb{R}

.
(a) Find a Cartesian equation of the curve. (3 marks)
(b) State the range of valid

x

-values, and explain why the Cartesian equation alone describes more of the plane than the curve does. (2 marks)

●●●●●Level 55 marksStart

Parametric curves and trig identities

16 questions

LessonNot started

A curve has parametric equations

x = \cos t,\ y = \sin t

0 \le t < 2\pi

. Find a Cartesian equation of the curve and describe it.

●●●●●Level 22 marksStart

A curve has parametric equations

x = 4\cos t,\ y = 4\sin t

. Find a Cartesian equation of the curve and state its centre and radius.

●●●●●Level 22 marksStart

A curve has parametric equations

x = 5\sin t, \; y = 5\cos t

\; 0 \leqslant t < 2\pi

.

Find a Cartesian equation of the curve. (2 marks)

●●●●●Level 22 marksStart

The diagram shows a curve

C

with parametric equations

x = 5\cos t,\ y = 2\sin t

0 \le t < 2\pi

.Find a Cartesian equation of

C

●●●●●Level 32 marksStart

A curve has parametric equations

x = 2 + 3\cos t,\ y = 1 + 3\sin t

. Find a Cartesian equation of the curve and state its centre and radius.

●●●●●Level 32 marksStart

A curve has parametric equations

x = 4\cos t, \; y = 3\sin t

\; 0 \leqslant t < 2\pi

.
(a) Find a Cartesian equation of the curve. (3 marks)
(b) State the range of possible values of

x

. (1 mark)

●●●●●Level 34 marksStart

A curve has parametric equations

x = 1 + 2\cos t, \; y = -3 + 2\sin t

\; 0 \leqslant t < 2\pi

.

Find a Cartesian equation of the curve, and state what the curve is. (4 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = 3\cos t,\ y = 2\sin t

.
(a) Find a Cartesian equation of the curve.
(b) State the range of values of

x

and of

y

●●●●●Level 44 marksStart

A curve has parametric equations

x = \sec t,\ y = \tan t

. Show that a Cartesian equation of the curve is

x^2 - y^2 = 1

●●●●●Level 42 marksStart

A curve has parametric equations

x = 2\sin t,\ y = \cos 2t

. Find a Cartesian equation of the curve in the form

y = f(x)

, and state the range of values of

x

●●●●●Level 43 marksStart

A curve has parametric equations

x = 3\sin t, \; y = \cos 2t

.

Find a Cartesian equation of the curve. (4 marks)

●●●●●Level 44 marksStart

A curve has parametric equations

x = 2\tan t, \; y = \sec^{2} t

.

Find a Cartesian equation of the curve. (4 marks)

●●●●●Level 44 marksStart

A curve has parametric equations

x = \cos t, \; y = \cos 2t

\; 0 \leqslant t \leqslant \pi

.
(a) Show that a Cartesian equation of the curve is

y = 2x^{2} - 1

. (3 marks)
(b) State the range of values of

x

for which this equation describes the curve. (1 mark)

●●●●●Level 44 marksStart

The curve

C

has parametric equations

x = \tan t + 5 \qquad\qquad y = \sec^2 t + 2 \qquad\qquad -\frac{\pi}{2} < t < \frac{\pi}{4}

The line

l

is the normal to

C

at the point

P

where

t = -\dfrac{\pi}{4}

(a) Using parametric differentiation, show that an equation for

l

y = \tfrac{1}{2}x + 2

(5 marks)
(b) Show that all points on

C

satisfy the equation

y = (x - 5)^2 + 3

(2 marks)

The straight line with equation

y = \tfrac{1}{2}x + k

, where

k

is a constant, intersects

C

at two distinct points.
(c) Find the range of possible values for

k

.

(5 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = \cos t,\ y = \sin 2t

. Show that a Cartesian equation of the curve is

y^2 = 4x^2(1 - x^2)

●●●●●Level 53 marksStart

A curve has parametric equations

x = \sin t, \; y = \sin\left(t + \dfrac{\pi}{3}\right)

for

0 \leqslant t \leqslant \dfrac{\pi}{2}

.

Show that a Cartesian equation of the curve is

y = \dfrac{x}{2} + \dfrac{\sqrt{3}}{2}\sqrt{1 - x^{2}}

(5 marks)

●●●●●Level 55 marksStart

Intersections of parametric curves

19 questions

LessonNot started

A curve has parametric equations

x = t + 3,\ y = t^2 - 4

.
(a) Find the coordinates of the points where the curve meets the

x

-axis.
(b) Find the coordinates of the point where the curve meets the

y

-axis.

●●●●●Level 23 marksStart

A curve has parametric equations

x = 2t,\ y = t^2 - 2t - 3

. Find the coordinates of the points where the curve crosses the

x

-axis.

●●●●●Level 33 marksStart

A curve has parametric equations

x = t^2,\ y = 2t

. Find the coordinates of the points where the curve meets the line

y = x - 3

●●●●●Level 33 marksStart

A curve has parametric equations

x = t + 1,\ y = t^2

. Find the coordinates of the points where the curve meets the line

y = x + 1

●●●●●Level 33 marksStart

The line

y = 3x - 4

meets the curve with parametric equations

x = 2t,\ y = t^2 + 1

. Find the coordinates of the two points of intersection.

●●●●●Level 33 marksStart

A curve has parametric equations

x = t^2 - 1,\ y = 2t

.
(a) Find the coordinates of the point where the curve meets the

x

-axis.
(b) Find the coordinates of the points where the curve meets the

y

-axis.

●●●●●Level 34 marksStart

A curve

C

has parametric equations

x = t^2 - 4,\ y = t^3 - 4t

.
(a) Find the coordinates of the points where

C

meets the

x

-axis.
(b) Show that

C

meets the

y

-axis at only one point, and explain why

C

passes through that point twice.

●●●●●Level 35 marksStart

A curve has parametric equations

x = 2t,\ y = t^2

. Show that the line

y = x - 1

is a tangent to the curve, and find the coordinates of the point of contact.

●●●●●Level 33 marksStart

A curve has parametric equations

x = t + 2, \; y = t^{2} - 1

.

Find the coordinates of the points where the curve crosses the

x

-axis. (3 marks)

●●●●●Level 33 marksStart

A curve has parametric equations

x = t + 2, \; y = t^{2} + 2t

.

Find the coordinates of the points where the curve meets the line

y = 8

. (4 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = 2t, \; y = t^{2} - 3

.

Find the coordinates of the points where the curve meets the line

x + y = 5

. (5 marks)

●●●●●Level 35 marksStart

A curve has parametric equations

x = t^{2}, \; y = t^{3}

.

Find the coordinates of the points where the curve meets the line

y = 2x

. (4 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = t^{2}, \; y = 2t

.

Show that the line

y = x + 1

is a tangent to the curve, and find the coordinates of the point of contact. (5 marks)

●●●●●Level 35 marksStart

The points

P

and

Q

on the curve

x = t^{2}, \; y = 2t

correspond to

t = 1

and

t = 3

.

Find an equation of the chord

PQ

, giving your answer in the form

ax + by + c = 0

with integer coefficients. (4 marks)

●●●●●Level 34 marksStart

A curve has parametric equations

x = t^{2} - 9, \; y = t + 2

.

The curve crosses the

y

-axis at the points

A

and

B

. Find the length

AB

. (4 marks)

●●●●●Level 34 marksStart

The curve

C

has parametric equations

x = t^3 + 2 \qquad\qquad y = 9 - t^2 \qquad\qquad -3 \leqslant t \leqslant 3

(a) Find the coordinates of the two points where

C

meets the

x

-axis.

(2 marks)

The finite region

R

is bounded by

C

and the

x

-axis.
(b) Using algebraic integration, find the exact area of

R

.

(4 marks)

●●●●●Level 46 marksStart

The curve

C

has parametric equations

x = \tan t + 5 \qquad\qquad y = \sec^2 t + 2 \qquad\qquad -\frac{\pi}{2} < t < \frac{\pi}{4}

The line

l

is the normal to

C

at the point

P

where

t = -\dfrac{\pi}{4}

(a) Using parametric differentiation, show that an equation for

l

y = \tfrac{1}{2}x + 2

(5 marks)
(b) Show that all points on

C

satisfy the equation

y = (x - 5)^2 + 3

(2 marks)

The straight line with equation

y = \tfrac{1}{2}x + k

, where

k

is a constant, intersects

C

at two distinct points.
(c) Find the range of possible values for

k

.

(5 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = \cos 2t \qquad y = \sin t\cos 2t \qquad -\dfrac{\pi}{2} < t < \dfrac{\pi}{2}

where

t

is a parameter.

The point

P

lies on

C

where

t = \dfrac{\pi}{6}

The line

l

, shown in Figure 1, is the tangent to

C

P

.
(a) Use parametric differentiation to show that

(i)

\dfrac{dy}{dx} = p\sin t + q\operatorname{cosec} t

where

p

and

q

are constants to be found

(ii) an equation for

l

2x - 8y + 1 = 0

(6 marks)

The line

l

intersects the curve

C

again at the point

Q

.
(b) Using algebra and showing detailed reasoning, find the exact coordinates of

Q

.

(6 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = t^2 + 1, \qquad y = t^3 - 4t,

where

t

takes all real values.
(a) The curve meets the

x

-axis at the points

A

and

B

, where

A

is closer to the origin. Find the coordinates of

A

and

B

, and show that the curve passes through

B

for two distinct values of

t

. (3 marks)
(b) The curve has a loop. Determine the exact area of the region enclosed by the loop. (5 marks)

●●●●●Level 58 marksStart

Modelling with parametric equations

21 questions

LessonNot started

A ball is thrown so that,

t

seconds after release, its position in metres is given by

x = 20t,\ y = 15t - 5t^2

. Its path is shown below.(a) Find the position of the ball after

1

second.
(b) Find the time at which the ball lands.
(c) Find the horizontal distance travelled before it lands.

●●●●●Level 34 marksStart

A ball is launched from ground level so that,

t

seconds after launch, its position in metres is

x = 8t,\ y = 6t - 5t^2

.
(a) Find the greatest height reached and the time at which it occurs.
(b) Find the horizontal range (the distance from launch to where it lands).

●●●●●Level 35 marksStart

A particle moves so that, at time

t

, its position is

x = 10t,\ y = 10t - 5t^2

. Find the Cartesian equation of the path of the particle.

●●●●●Level 32 marksStart

A particle moves so that,

t

seconds after it starts (

t \ge 0

), its position in metres is

x = 3t,\ y = 4t

.
(a) Show that the path is a straight line through the origin, and find its gradient.
(b) Find the distance of the particle from the origin when

t = 2

●●●●●Level 34 marksStart

A stone is thrown so that,

t

seconds after release (

t \ge 0

), its position in metres is

x = 2 + 3t,\ y = 1 + 4t - 5t^2

.
(a) Write down the position from which the stone is thrown.
(b) Find the greatest height of the stone and the time at which it occurs.
(c) Find the Cartesian equation of the path.

●●●●●Level 35 marksStart

A rider sits on a Ferris wheel. Her position in metres at time

t

(in radians,

t \ge 0

) is modelled by

x = 8\cos t,\ y = 10 + 8\sin t

.
(a) State the centre and radius of the wheel.
(b) Find the rider's maximum and minimum height.
(c) Find the Cartesian equation of her path.

●●●●●Level 35 marksStart

A stone is thrown from the top of a cliff. Its position in metres at time

t

seconds (

t \ge 0

) is modelled by

x = 15t,\ y = 20 + 10t - 5t^2

, where

y

is the height above the sea.
(a) State the height of the cliff.
(b) Find the exact time at which the stone hits the sea.
(c) Find the horizontal distance from the cliff to the point where it lands, to 3 significant figures.

●●●●●Level 36 marksStart

A boat's position in kilometres,

t

hours after noon (

t \ge 0

), is modelled by

x = 2t,\ y = t^2 - 4t

, where the origin is a harbour, the

x

-axis points east and the

y

-axis points north.
(a) Find the boat's position at

t = 0

and at

t = 4

.
(b) Show that the boat returns to the line

y = 0

(due east of the harbour) and find when.
(c) Find the most southerly point the boat reaches and the time at which it does.

●●●●●Level 36 marksStart

A ball is kicked from ground level. Its position in metres,

t

seconds after being kicked, is modelled by

x = 15t, \qquad y = 20t - 5t^{2}, \qquad t \geqslant 0

(a) Find the time at which the ball lands. (2 marks)
(b) Find the horizontal distance the ball travels before landing. (2 marks)

●●●●●Level 34 marksStart

A projectile's height in metres,

t

seconds after launch, is modelled by

y = 20t - 5t^{2}

.

Find the maximum height reached. (3 marks)

●●●●●Level 33 marksStart

A drone flies in a horizontal circle. Its position in metres at time

t

is modelled by

x = 5 + 3\cos t, \qquad y = 4 + 3\sin t

(a) Write down the centre and radius of the circular path. (2 marks)
(b) Find a Cartesian equation of the path. (2 marks)

●●●●●Level 34 marksStart

A stone is thrown horizontally from the top of a cliff. Its position in metres,

t

seconds after release, is modelled by

x = 12t, \qquad y = 45 - 5t^{2}, \qquad t \geqslant 0

where

y

is the height above the sea.
(a) Write down the height of the cliff. (1 mark)
(b) Find the time when the stone hits the sea. (2 marks)
(c) Find the horizontal distance from the cliff at that moment. (2 marks)

●●●●●Level 35 marksStart

A particle moves so that,

t

seconds after it starts, its position is

x = 8t, \qquad y = 6t - t^{2}

Find the height

y

of the particle at the moment when

x = 24

. (3 marks)

●●●●●Level 33 marksStart

A ball is kicked from ground level towards a wall. Its position in metres,

t

seconds after the kick, is modelled by

x = 10t, \qquad y = 15t - 5t^{2}

The wall is

20

m away and

8

m tall. Determine, with working, whether the ball passes over the wall, and by how much it clears or falls short. (5 marks)

●●●●●Level 35 marksStart

A boat leaves a harbour at noon. Its position in kilometres,

t

hours after noon, is modelled by

x = 3t, \qquad y = 4t, \qquad t \geqslant 0

where the harbour is at the origin.
(a) Show that the boat's distance from the harbour at time

t

5t

km. (2 marks)
(b) State the boat's speed. (1 mark)
(c) Find the time of day at which the boat is

20

km from the harbour. (2 marks)

●●●●●Level 35 marksStart

A designer is making a curved metal bracket to hold a hanging flower basket.

The edge of the bracket can be modelled by the parametric equations

x = 10t - \frac{5}{t} \qquad y = 3t^2 + \frac{12}{t}

for

1 \leq t \leq 3

, as shown in the diagram below, where

x

and

y

are measured in centimetres.Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(3 marks)

●●●●●Level 33 marksStart

A particle moves in the

x

y

plane so that at time

t

seconds, where

t \ge 0

, its coordinates are given by

x = e^{2t} - 7e^{t} + 10, \qquad y = e^{t} + 4e^{-t}.

(a) Explain why the path of the particle never crosses the

x

-axis.

(1 mark)
(b) Determine the exact values of

t

when the path of the particle intersects the

y

-axis.

(2 marks)
(c) Show that

\dfrac{dy}{dx} = \dfrac{e^{2t} - 4}{e^{2t}(2e^{t} - 7)}

.

(4 marks)
(d) Hence find the coordinates of the particle when its path is parallel to the

y

-axis.

(3 marks)

●●●●●Level 410 marksStart

A craftsperson makes an arch-shaped wooden panel. The panel is modelled by the region between the

x

-axis and the curve with parametric equations

x = k(2\theta - \sin 2\theta), \qquad y = 10\sin^2\theta,

where

k

is a positive constant and

0 \le \theta \le \pi

. The diagram shows the curve. Lengths are in centimetres and the area of the panel must be

180\text{ cm}^2

.
(a) Show that the area of the panel is given by

\displaystyle 40k\int_0^{\pi} \sin^4\theta \, d\theta

.

(3 marks)
(b) Determine the exact value of

k

.

(6 marks)

●●●●●Level 49 marksStart

A design for a wooden serving platter is shown in Figure 1.

Figure 1The platter is symmetrical about its length.

The curve of the top half of the platter can be modelled by the parametric equations

x = 5t^2 + 6t \qquad y = 9t - t^3

for

0 \leq t \leq 3

, as shown in Figure 2, where

x

and

y

are measured in centimetres.

Figure 2(a) Find the length of the platter.

(2 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) Hence, show that the width of the platter is approximately one third of its length.

(7 marks)

●●●●●Level 49 marksStart

A furniture maker is designing a rocking chair, shown in Figure 1.

Figure 1The curve of each runner of the rocking chair can be modelled by the parametric equations

x = 20t \qquad y = 2t^2 + \frac{32}{t}

for

1 \leq t \leq 4

, as shown in Figure 2, where

x

and

y

are measured in centimetres.

P

is the point where

t = 1

and

Q

is the point where

t = 4

Figure 2(a) Show that a Cartesian equation of the curve is

y = \frac{x^2}{200} + \frac{640}{x}

(2 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) The chair rests on horizontal ground, with each runner touching the ground at its lowest point.

Hence, find the height of

Q

above the ground.

(6 marks)
(c) Suggest one limitation of the model.

(1 mark)

●●●●●Level 49 marksStart

A company that builds adventure playgrounds is designing a rope bridge between two platforms, as shown in Figure 1.

Figure 1The walkway of the bridge can be modelled by the parametric equations

x = t^2 + 2t \qquad y = 0.2t^3 - 1.6t + 6

for

0 \leq t \leq 3

, as shown in Figure 2, where

x

and

y

are measured in metres and the

x

-axis represents the horizontal ground.

P

is the point where

t = 0

and

Q

is the point where

t = 3

Figure 2(a) So that children can step on and off the bridge safely, the difference in height between

P

and

Q

must be less than

1

metre.

Show that the bridge meets this safety requirement.

(3 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) A vertical support,

RS

, is to be added between the ground and the lowest point on the bridge, as shown in Figure 3 below.

Figure 3Find the length of

RS

(iii) Find the acute angle the bridge makes with the horizontal at

Q

Give your answer to the nearest degree.

(9 marks)

●●●●●Level 512 marksStart