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°.

Start practising Read the lesson

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
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 line with equation $x + 3y - 13 = 0$ is the tangent to $C$ at the point $P$ .
(b) Find the coordinates of $P$ .

(4 marks)
(c) Find an equation of the normal to $C$ at $P$ . Give your answer in the form $y = mx + c$ , where $m$ and $c$ are integers.

(3 marks)
Level 4 · 10 marksTry it
2
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 $L$ , with equation $y = mx - 4$ where $m$ is a constant, touches $C$ at the point $P$ .
(b) Show that

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

(3 marks)
(c) Hence find the coordinates of each of the two possible positions of $P$ .

(4 marks)
Level 4 · 10 marksTry it
3
The equation of a circle is $x^{2} + y^{2} + 6x - 8y - 25 = 0$ .
(a) State the coordinates of the centre of the circle.

(1 mark)

The points $P$ and $Q$ lie on the circle. The midpoint of the chord $PQ$ is $M(1, 7)$ .
(b) Find the equation of the chord $PQ$ , giving your answer in the form $y = mx + c$ .

(4 marks)
(c) Find, by calculation, the $x$ -coordinates of $P$ and $Q$ .

(3 marks)
Level 4 · 8 marksTry it
4
The curve $C$ has equation $x^2 + y^2 + 10x - 4y = k$ , where $k$ is a constant.
(a) Find the set of values of $k$ for which $C$ is a circle.

(2 marks)

In the case $k = 11$ , the tangents to $C$ at the points $A$ and $B$ both have gradient $3$ .
(b) Determine the coordinates of $A$ and $B$ .

(5 marks)
Level 4 · 7 marksTry it
5
The points $A(-2, 6)$ , $B(3, 7)$ and $C(10, 0)$ all lie on a circle.
(a) Find the equation of the perpendicular bisector of $AB$ , giving your answer in the form $y = mx + c$ .

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

Hence, or otherwise, find the equation of this circle.

(5 marks)
Level 4 · 8 marksTry it