A-level · Coordinate geometry

Lines and circles — tangents, chords, and intersectionsExam-style questions

Where a line meets a circle (substitute, then read the discriminant), and the three circle facts that solve coordinate problems: tangent ⊥ radius, the perpendicular bisector of a chord passes through the centre, and the angle in a semicircle is 90°.

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20 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
In this question you must show detailed reasoning.

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

Figure 1 shows part of the curve $C_1$ with equation

$y = 1 + 3x^{2} - x^{3}$

together with the circle $C_2$ . The centre of $C_2$ is the point $T$ , where $T$ is the maximum turning point of $C_1$
(a) Use calculus to find the coordinates of $T$ .

(3 marks)

The circle $C_2$ passes through the point $A$ on $C_1$ at which $x = 3$
(b) Find an equation for $C_2$ .

(3 marks)

The line $l$ touches $C_2$ at $A$ .
(c) Show that $l$ has equation

$x - 4y + 1 = 0$

(3 marks)

The $y$ -axis, the line $l$ and the curve $C_1$ enclose the region $R$ , shown shaded in Figure 1.
(d) Use algebraic integration to find the exact area of $R$ .

(3 marks)
Level 5 · 12 marksTry it
2
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 5 · 7 marksTry it
3
The line $L_1$ has equation $x + 2y = 4$ and the line $L_2$ has equation $x + 2y = 24$

$L_1$ crosses the $x$ -axis at the point $P$

The line through $P$ perpendicular to $L_1$ meets $L_2$ at the point $Q$ , as shown in the diagram.(a) (i) Find the coordinates of $Q$

(ii) Hence find the shortest distance between $L_1$ and $L_2$

Give your answer in the form $k\sqrt{5}$ , where $k$ is an integer.

(7 marks)
(b) The circle $C$ touches both $L_1$ and $L_2$ , and the centre of $C$ lies on the $y$ -axis.
(b) (i) Find the coordinates of the centre of $C$

(ii) Find the equation of $C$

(4 marks)
Level 5 · 11 marksTry it
4
A circle has equation $(x - 4)^{2} + (y + 1)^{2} = 40$ . A straight line has gradient $m$ and passes through the point $(0, 7)$ . The line is a tangent to the circle.

Find the two possible values of $m$ . For each value of $m$ , find the coordinates of the point at which the line touches the circle.

(8 marks)
Level 5 · 8 marksTry it
5
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 5 · 10 marksTry it

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