Browse Coordinate geometry

176 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Straight lines

23 questions

LessonNot started

Kofi is comparing the cost of hiring an electric bike from two companies. Company A has a fixed charge of £3 and then charges 12p per minute. Company B has no fixed charge, does not charge for the first 20 minutes and then charges 18p per minute.
(a) Find an expression for the cost, in pounds, of hiring from each of the companies in terms of

m

m

is the number of minutes of hire and

m > 20

.

(2 marks)
(b) Hence find the value of

m

for which both companies would charge the same.

(2 marks)

●●●●●Level 24 marksStart

The depth of water in a lake was monitored.

Exactly 3 years after monitoring began, the depth of water was 42.6 metres.

Exactly 8 years after monitoring began, the depth of water was 38.1 metres.

Given that the depth of water,

D

t

years after monitoring began can be modelled by the equation

D = a + bt

a

b

are constants,
(a) find the value of

a

and the value of

b

.

(3 marks)

According to the model, exactly

T

years after monitoring began, the depth of water was 31.5 metres.

Making your method clear,
(b) find the value of

T

, giving your answer to one decimal place.

(Solutions relying entirely on calculator technology are not acceptable.)

(2 marks)

●●●●●Level 25 marksStart

The number of visitors to a science museum each year is monitored.

Exactly 2 years after monitoring began, the number of visitors that year was 14 600

Exactly 6 years after monitoring began, the number of visitors that year was 23 000

The number of visitors each year,

N

thousand, is modelled by the equation

N = pt + q

t

is the number of years after monitoring began and

p

q

are constants.
(a) Find the value of

p

and the value of

q

.

(3 marks)

Exactly

T

years after monitoring began, the number of visitors that year was 40 000
(b) Find the value of

T

, according to the model, giving your answer to one decimal place.

(2 marks)

●●●●●Level 25 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

a

b

c

are integers. (3 marks)

●●●●●Level 25 marksStart

C

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

k

is a constant.

Find, in terms of

k

,

(i) the coordinates of the centre of

C

(ii) the radius of

C

●●●●●Level 23 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

a

b

c

are integers. (3 marks)

●●●●●Level 25 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

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

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

●●●●●Level 23 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

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

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

●●●●●Level 23 marksStart