Equation of a circle

Example 1

Find all points whose distance from the origin is 3.

Points on axes

Other points

All points

The points at a distance of 3 from the origin form a circle with radius of 3. We denote the point (x, y) on the circle and find the distance of this point from the origin using the formula for the distance between the two points. This is how we get the equation for the set of points that represents the circle.

This is an origin-centred circle with a radius of 3.

The equation of a circle with a centre point

Let us denote the centre of a circle (x₀, y₀). The distance of the point (x, y) from the centre is

Let this distance, i.e. the radius of the circle, be denoted by the letter r

This is the equation of a circle with a centre point

From this, we can easily find the centre point and radius

Example 2

Centre (4,2), radius 5

Centre (-5,1), radius 2

Centre (-7,0), radius 10

The general form equation of a circle

Usually a circle comes in a general form of the equation. Then it has no parentheses and all the terms are to the left of the equation.

We change the following equation of the circle from the centre point form to the general form.

Example 3

From a general form equation of a circle, neither the centre nor the radius can be seen directly. In this case, one must go to the other direction and form a centre point form equation. Here helps to remember the formula for the binomial square.

Example 4

Below is the general form equation of a circle

Let's change the equation to the centre point form. We group x and y and take the constants to the right of the equation. Then we add numbers to the equation to form binomial squares.

The centre of the circle is (-2,4) and the radius is 3

Turn on the subtitles if needed