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 circle $C_1$ has equation

$x^{2} + y^{2} - 10x + 2y + 2 = 0$
(a) Find

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

(ii) the exact radius of $C_1$ .

(3 marks)

In part (b) you must show detailed reasoning.

A second circle, $C_2$ , has equation

$(x + 2)^{2} + (y - 2)^{2} = 6$
(b) Prove that $C_1$ and $C_2$ have no points in common.

(3 marks)
Level 3 · 6 marksTry it
2
The line with equation $y = 3x + c$ , where $c$ is a constant, is a tangent to the circle with equation

$x^2 + y^2 = 10$

Find the two possible values of $c$ . (5 marks)
Level 3 · 5 marksTry it
3
The line segment joining $P(-1, 7)$ and $Q(5, -3)$ is a diameter of a circle.

Determine the equation of this circle in the form $x^2 + y^2 + ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers to be determined.

(4 marks)
Level 3 · 4 marksTry it
4
A circle $C$ has equation

$x^2 + y^2 - 6x + 12y - 4 = 0$
(a) Find

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

(ii) the radius of $C$

(3 marks)

The origin is $O$ . The point $P$ on $C$ is such that the distance $OP$ is a maximum.
(b) Find the exact length $OP$ .

(2 marks)
Level 3 · 5 marksTry it
5
The circle $C$ has equation

$x^2 + y^2 + 14x - 8y + 40 = 0$
(a) Find

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

(ii) the radius of $C$ .

(3 marks)

The point $Q$ lies on $C$ and is nearer to the origin $O$ than any other point of $C$ .
(b) Find the exact length $OQ$ .

(2 marks)
Level 3 · 5 marksTry it