For individual or collaborative investigation.(exercise)The distance formula, the midpoint...

For individual or collaborative investigation.

(exercise)

The distance formula, the midpoint formula, and the 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.

Work Exercise in order to see the relationships among these concepts.

Exercise

Use the method described in Exercises 1–2 to find the center-radius form of the equation of the circle with diameter having endpoints (3,-5) and (-7, 3).

Exercise 1

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 2

Using the center found in Exercise 3 and the radius found in Exercises 4–5, give the center-radius form of the equation of the circle.

Exercise 3

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 4

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 your result from Exercise 3 and the distance formula to find the radius.

Exercise 5

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. You should once again obtain the same answer you found in Exercise 4.