__Category__: Analytic Geometry, Plane Geometry, Algebra"Published in Vacaville, California, USA"

Find the equation of a circle that passes through the points of intersection of the circles x

^{2}+ y

^{2}= 2x and x

^{2}+ y

^{2}= 2y, and has its center on the line y = 2.

__Solution__:

To illustrate the problem, it is better to draw the figure as follows

A circle that passes through the points of intersection of two circles and its center at y = 2. (Photo by Math Principles in Everyday Life) |

The equation of a chord or radical axis can be solved by subtracting the equations of two circles, as follows

Substitute y = x to either of the equations of a circle, we have

After equating each factor to zero, the values of x are 0 and 1. Hence, the points of intersection of two circles are (0, 0), and (1, 1).

The center of a circle can be solved by using the distance of two points formula as follows

Hence, the center of a circle is C (-1, 2). The radius of a circle is

Therefore, the equation of a circle is