Circle: Three points defining a right angle: Find the equation of the circle on which
the points (-2, 2), (4, 2), and (4, -6) lie.

Upon plotting these points we see that they define a right triangle. For a circle passing through the points, the hypotenuse is the diameter. The distance between (-2, 2) and (4, -6) must be the length of the diameter, or twice the radius.

From the definition of distance, we get

Hence the radius is 5. Next we need to locate the center of the circle, which is at the midpoint of this diameter. Using the definition of the midpoint of the line joining two points, we get for the coordinates:

, or .

Hence h = 1, k = -2, and the equation of the circle with the coordinates of the center as h and k: h = 1, k = -2:

We can check this equation by substituting the coordinates of the third point (4, 2):

, and the equation is correct.