# Equation of a Circle with Endpoints of Diameter

Let and be the end points of the diameter of the circle as shown in the diagram.

Let be any point of the circle. Connecting the points and with the point makes an angle between them. First we find the slopes of the lines and as:

Slope of the line

Slope of the line

Since , the lines and are perpendicular to each other. Therefore, the product of their slopes is . i.e.:

This is the equation of the circle through the extremities (ends) of its diameter. In order to find the centre and radius of this circle, we simplify the above equation of a circle as follows:

Comparing this equation with the general equation of a circle, we have

Therefore, the centre of the circle is given by

The radius of the circle is given by

__Example__**:**

Find the equation of a circle through the ends and of its diameter. Also find the centre and radius.

The equation of the circle through the ends points of its diameter is

Here from the given points we have values

Now substitute these values of the given points in the above equation of a circle as

The centre of the circle is

The radius of the circle is