# Equation of Circle with Endpoints of Diameter

Let $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$ be the end points of the diameter of the circle as shown in the diagram.

Let $P\left( {x,y} \right)$ be any point of the circle. Connecting the points $A$ and $B$ with the point $P$ and makes an angle ${90^ \circ }$ between them. First we find the slopes of the lines $PA$ and $PB$ as:
Slope of the line $PA = \frac{{y - {y_1}}}{{x - {x_1}}}$
Slope of the line $PB = \frac{{y - {y_2}}}{{x - {x_2}}}$
Since $m\angle APB = {90^ \circ }$, so the lines $PA$ and $PB$ are perpendicular to each other, therefore, the product of their slopes is $- 1$. 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 above equation of circle as follows:

Comparing this equation with the general equation of 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 circle through the ends $\left( {5,7} \right)$ and $\left( {1,3} \right)$ of its diameter. Also find the centre and radius.
Equation of circles through ends points of its diameter is

Here from the given points we have values ${x_1} = 5,\,\,{x_2} = 1,\,\,{y_1} = 7,\,\,{y_2} = 3$
Now substitute these values of the given points in the above equation of circle as

The centre of the circle is $\left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right) = \left( {\frac{{5 + 1}}{2},\frac{{7 + 3}}{2}} \right) = \left( {3,5} \right)$