A circle passing through two point and having its center on the line. An equation of circle having the join of A(x₁, y₁) and B(x₂, y₂) as a diameter

Since line AB is a diameter of the circle. Its midpoint is the center of the circle. The radius of the circle is known, as the standard form of an equation of the circle may be easily written. However, a more elegant procedure is to make use of the plane geometry. If P(x, y) is any point on the circle,

then

m ∠ APB=90

Thus, the line AP and BP are perpendicular to each other.

and

When line are perpendicular to each other, the resulting value is negative

After simplification, we can be written as

This is the equation of a circle passing through two point, having its center on the line.

Example:

A circle passing through two point and having tangent at one of these points is known.

For example, equation of the circle passing through the point (-2, -5) and touching the line 3x+4y-24=0