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
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 3 · 5 marksTry it
2
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 3 · 4 marksTry it
3
Find the length of the tangent from the point $P(10, 4)$ to the circle $(x - 2)^2 + (y - 1)^2 = 9$ .
Level 3 · 3 marksTry it
4
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 $K$ of $C$ lies on the line $l$ with equation $y = 2x - 7$ , and the $x$ -coordinate of $K$ is $p$ , where $p > 0$ .
(a) Explain why the radius of $C$ is $p$ . (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 3 · 9 marksTry it
5
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 3 · 4 marksTry it

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