The equation of a circle describes every point that lies on its circumference using its coordinates. It allows you to calculate the centre and radius of a circle, write an equation from given information, and solve problems involving coordinate geometry.

(x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2

where ((a,b)) represents the centre of the circle and (r) is its radius.

In this guide, we will explore the main circle equation formulae, explain how to identify the centre and radius, and demonstrate how to rearrange an equation by completing the square.

1

Convert the circle equation from general form to centre-radius form and state the centre and radius: .

Solution

Group and complete the square:

.

So the centre is and the radius is .

2

A circle has centre and passes through the point .

Write its equation in the form .

Solution

Radius squared is the squared distance from centre to the point: .

Equation: .

3

Find the centre and radius of the circle:

.

Solution

The equation is already in centre-radius form .

So centre and radius .

4

Determine whether the point lies inside, on, or outside the circle .

Solution

Centre is and . Compute squared distance from centre to the point: .

Since , the point lies inside the circle.

5

Find the equation of the circle with diameter endpoints and , giving your answer in centre-radius form.

Solution

The centre is the midpoint of the diameter: .

The diameter length squared is , so radius squared is a quarter of that: .

Equation: .

6

A circle has equation and radius . Find the value of .

Solution

Complete the square:

.

Thus . Given , , so .

7

Find the equation of the circle centred at the origin that passes through .

Give your answer in the form .

Solution

With centre , we have .

So the equation is .

8

Two circles are given by and .

Calculate the distance between their centres and decide whether the circles intersect, touch, or are separate.

Solution

Centres are and .

Distance between centres: .

Radii are and .

Compute and .

Since , the circles intersect at two points.

9

The circle is translated by the vector .

Write the equation of the translated circle in centre-radius form.

Solution

First find the original centre and radius by completing the square:

.

Original centre is , radius squared .

Translate the centre by : new centre .

Radius unchanged. Translated circle: .

10

A circle has centre . It is tangent to the y-axis (the line ). Write the equation of the circle in centre-radius form.

Solution

If the circle is tangent to , the radius equals the perpendicular distance from the centre to the y-axis.

Distance from to is , so .

Equation: .

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Emma

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.