Coordinate geometry

Equations of straight lines, the equation of a circle in both forms, and the circle properties used in proofs.

9 min read

Coordinate geometry connects algebra and shape: lines, circles, and the standard circle properties. Knowing the circle equation in both forms is essential.

A line can be written as $y - y_{1} = m (x - x_{1})$ (point–gradient form) or $a x + b y + c = 0$ . Parallel lines share a gradient; perpendicular gradients satisfy $m_{1} m_{2} = - 1$ .

Worked example

Find the line through $(2, - 1)$ and $(6, 7)$ .

Gradient $m = \frac{7 - ( - 1 )}{6 - 2} = 2$ . Then $y + 1 = 2 (x - 2)$ , i.e. $y = 2 x - 5.$

A circle drawn on x-y axes with centre at the point (a, b) and a radius line of length r marked to the circumference. — A circle of centre (a, b) and radius r has equation (x − a)² + (y − b)² = r².

A circle with centre $(a, b)$ and radius $r$ is $(x - a)^{2} + (y - b)^{2} = r^{2}$ . Expanded, it takes the form $x^{2} + y^{2} + c x + d y + e = 0$ ; complete the square in $x$ and $y$ to recover the centre and radius.

Worked example

Find the centre and radius of $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ .

$(x - 3)^{2} - 9 + (y + 2)^{2} - 4 - 12 = 0 \Rightarrow (x - 3)^{2} + (y + 2)^{2} = 25.$

Centre $(3, - 2)$ , radius $5$ .

The properties used in proofs:

The perpendicular from the centre to a chord bisects the chord.
A tangent is perpendicular to the radius at the point of contact.
The angle at the centre is twice the angle at the circumference on the same arc.
The angle in a semicircle is $90°$ .
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral sum to $180°$ .
The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Exam tip

"Tangent ⊥ radius" turns many circle problems into a right-angled triangle you can attack with Pythagoras, so look for it whenever a tangent appears.

Reading the centre as $(- a, - b)$ instead of $(a, b)$ from $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
Forgetting that the radius is $r^{2}$ , not $r^{2}$ , after completing the square.
Using $m_{1} = m_{2}$ for perpendicular lines instead of $m_{1} m_{2} = - 1$ .

Coordinate geometry

Overview

Equations of straight lines

The equation of a circle

Circle properties

Common mistakes