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
A circle passes through $A(1, 1)$ , $B(1, 7)$ and $C(7, 7)$ . Find its equation.
Level 5 · 4 marksTry it
2
The equation $x^2 + y^2 - 8x + ky + 9 = 0$ represents a circle of radius $4$ . Find the possible values of $k$ .
Level 5 · 3 marksTry it
3
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A circle $C$ has equation

$x^2 + y^2 + 4kx - 2ky + 15 = 0$

where $k$ is a constant.
(a) Find in terms of $k$ ,

(i) the coordinates of the centre of $C$

(ii) the radius of $C$

(3 marks)

The line $l$ has equation $y = 3x + 2$
(b) Show that the $x$ coordinates of any points of intersection of $l$ and $C$ satisfy

$10x^2 + (12 - 2k)x + 19 - 4k = 0$

(3 marks)

Given that $l$ and $C$ have two points of intersection,
(c) find the range of possible values of $k$ .

(4 marks)

Given also that $k$ is a negative integer,
(d) state the greatest possible value of $k$ .

(1 mark)
Level 5 · 11 marksTry it
4
The circle $C_1$ has equation

$x^2 + y^2 - 8x + 6y - 11 = 0$
(a) Find

(i) the coordinates of the centre of $C_1$

(ii) the radius of $C_1$

(3 marks)

A second circle, $C_2$ , has centre $(-2,\ 5)$ and radius $k$ , where $k$ is a positive constant.
(b) Show that the distance between the centre of $C_1$ and the centre of $C_2$ is $10$ .

(2 marks)
(c) Find the range of values of $k$ for which $C_1$ lies entirely inside $C_2$ , writing your answer in set notation.

(2 marks)

Given instead that $C_1$ and $C_2$ meet at exactly two points,
(d) find the range of values of $k$ , writing your answer in set notation.

(4 marks)
Level 5 · 11 marksTry it