The distance formula, midpoint formula, and center radius form of the equation of a circle are closely related in the following problem.

A circle has a diameter with endpoints (-1, 3) and (5, -9). Find the center-radius form of the equation of this circle.

Use the method described in Exercises 53–57 to find the center-radius form of the equation of the circle with diameter having endpoints (3, -5) and (-7, 3).

Exercise 53.

To find the center-radius form, we must find both  the radius and the coordinates of the center. Find the coordinates of the center  using the midpoint formula. (The center of the circle must be the midpoint of the  diameter.)

Exercise 54.

There are several ways to find the radius of the circle. One way is to find the distance between the center and the point (-1, 3). Use the result from Exercise 53 and the distance formula to find the radius.

Exercise 55.

Another way to find the radius is to repeat Exercise 54, but use the point (5, -9) rather than (-1, 3). Do this to obtain the same answer found in Exercise 54.

Exercise 56.

There is yet another way to find the radius. Because the radius is half the diameter, it can be found by finding half the length of the diameter. Using the endpoints of the diameter given in the problem, find the radius in this manner. The same answer found in Exercise 54 should be obtained.

Exercise 57.

Using the center found in Exercise 53 and the radius found in Exercises 54–56,  give the center-radius form of the equation of the circle.