220 CHAPTER 2 Graphs and Functions Relating Concepts For individual or collaborative investigation (Exercises 57–62) 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 1-1, 32 and 15, -92. Find the center-radius form of the equation of this circle. Work Exercises 57–62 in order, to see the relationships among these concepts. 57. 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.) 58. There are several ways to find the radius of the circle. One way is to find the distance between the center and the point 1-1, 32. Use the result from Exercise 57 and the distance formula to find the radius. 59. Another way to find the radius of the circle is to repeat Exercise 58, but use the point 15, -92 rather than 1-1, 32. Do this to obtain the same answer found in Exercise 58. 60. 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 58 should be obtained. 61. Using the center found in Exercise 57 and the radius found in Exercises 58–60, give the center-radius form of the equation of the circle. 62. Use the method described in Exercises 57–61 to find the center-radius form of the equation of the circle with diameter having endpoints 13, -52 and 1-7, 32. 52. Find the equation of the circle of least radius that contains the points 11, 42 and 1-3, 22 within or on its boundary. 53. Find all values of y such that the distance between 13, y2 and 1-2, 92 is 12. 54. Suppose that a circle is tangent to both axes, is in the third quadrant, and has radius 22. Find the center-radius form of its equation. 55. Find the shortest distance from the origin to the graph of the circle with equation x2 - 16x + y2 - 14y + 88 = 0. 56. Phlash Phelps, the morning radio personality on SiriusXM Satellite Radio’s Sixties on Six Decades channel, is an expert on U.S. geography. He loves traveling around the country to strange, out-of-the-way locations. The photo shows Phlash seated in front of a sign in a small Arizona settlement called Nothing. (Nothing is so small that it’s not named on current maps.) The sign indicates that Nothing is 50 mi from Wickenburg, AZ, 75 mi from Kingman, AZ, 105 mi from Phoenix, AZ, and 180 mi from Las Vegas, NV. Explain how the concepts of Example 6 can be used to locate Nothing, AZ, on a map of Arizona and southern Nevada. x y 0 3 –1 5 (–1, 3) (5, –9)
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