A-level · Coordinate geometry

The equation of a circleExam-style questions

Why (x − a)² + (y − b)² = r² is just the distance formula squared, reading the centre and radius off it, and completing the square to recover them from the expanded form x² + y² + 2gx + 2fy + c = 0.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The points $(-1, -4)$ and $(5, -10)$ are the ends of a diameter of the circle $C$ .
(a) Find an equation for $C$ .

(5 marks)
(b) Hence find, in exact form, the coordinates of the point on $C$ closest to the $x$ -axis.

(2 marks)
Level 4 · 7 marksTry it
2
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 4 · 6 marksTry it
3
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 $C$ at $P$ , giving your answer in the form $ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers. (5 marks)
Level 4 · 5 marksTry it
4
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 4 · 5 marksTry it
5
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 4 · 5 marksTry it

10 more questions on the equation of a circle

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