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Find the equation of a circle that passes through the points of intersection of the circles x2 + y2 = 2x and x2 + y2 = 2y, and has its center on the line y = 2.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| 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
