217 2.2 Circles The point 11, 22 lies on all three graphs. Thus, we can conclude that the epicenter of the earthquake is at 11, 22. S Now Try Exercise 43. EXAMPLE 6 Locating the Epicenter of an Earthquake Suppose receiving stations A, B, and C are located on a coordinate plane at the points 11, 42, 1-3, -12, and 15, 22. Let the distances from the earthquake epicenter to these stations be 2 units, 5 units, and 4 units, respectively. Where on the coordinate plane is the epicenter located? SOLUTION Graph the three circles as shown in Figure 18. From the graph it appears that the epicenter is located at 11, 22. To check this algebraically, determine the equation for each circle and substitute x = 1 and y = 2. A C B –5 5 –5 5 x y Figure 18 Station A: 1x - 122 + 1y - 422 = 22 Equation of a circle with center 11, 42 and radius 2 11 - 122 + 12 - 422 ≟4 Let x = 1 and y = 2. 0 + 4≟4 4 = 4 ✓ Station B: 1x + 322 + 1y + 122 = 52 Equation of a circle with center 1-3, -12 and radius 5 11 + 322 + 12 + 122 ≟25 Let x = 1 and y = 2. 16 + 9≟25 25 = 25 ✓ Station C: 1x - 522 + 1y - 222 = 42 Equation of a circle with center 15, 22 and radius 4 11 - 522 + 12 - 222 ≟16 Let x = 1 and y = 2. 16 + 0≟16 16 = 16 ✓ 2.2 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. The circle with equation x2 + y2 = 49 has center with coordinates and radius equal to . 2. The circle with center 13, 62 and radius 4 has equation . 3. The graph of 1x - 422 + 1y + 722 = 9 has center with coordinates . 4. The graph of x2 + 1y - 522 = 9 has center with coordinates .
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