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Circles

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  1. Circles 2.2 Center-Radius Form■General Form

  2. Center-Radius Form of a Circle • Find the center-radius form of the equation of each circle described. a)Center at (1, –2), radius 3 (h, k) = (1, –2), r = 3 b)Center at (0, 0), radius 2 (h, k) = (0, 0), r = 2

  3. Graphing Circles • Graph Identify the center and radius: (h, k) = (1, –2), r = 3

  4. Graphing Circles • Graph Identify the center and radius: (h, k) = (0, 0), r = 2

  5. Graphing Circles • Graph C: (h, k) = (0, 0), r = 5 Graph C: (h, k) = (–2, –3),

  6. Sec. 2.2 p. 198 (1-12 all, 13-16 all Part A only)Due on Monday, 13 January 2014. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. (a) Write the equation of the circle in center-radius form. (b) Graph the circle.

  7. College Algebra K/DCMonday, 13 January 2014 • OBJECTIVE TSW write the equation for a circle in (1) general form, and (2) use circles in a real-world application. • ASSIGNMENT DUE • Sec. 2.2 p. 198 (1-12 all, 13-16 all Part A only) wire basket • QUIZ: Sec. 2.1 is not graded.

  8. Circles 2.2 General Form ■ An Application

  9. Center-radius Form of a Circle • The center-radius form of a circle is • The center is • (h, k) • and the radius is • r. • Sometimes you are given the general form of a circle: • x2 + y2 + cx + dy + e = 0 • Just by looking at this equation, it’s hard to determine the center and radius. So, we use the technique of completing the square.

  10. Finding the Center and Radius by Completing the Square • Find the center and radius of Complete the square twice, once for x and once for y. Center: (–2, 4) Radius: 8

  11. Finding the Center and Radius by Completing the Square • Find the center and radius of Group the terms and factor:

  12. Finding the Center and Radius by Completing the Square Complete the squares on x and y, then factor:

  13. Finding the Center and Radius by Completing the Square Center: Radius: 5 Divide both sides by 2:

  14. Determining Whether a Graph is a Point or Nonexistent • The graph of the equation • is either a point or is nonexistent. Which is it? Complete the square twice, once for x and once for y.

  15. Determining Whether a Graph is a Point or Nonexistent Since –2 < 0, the graph of is nonexistent. Factor. If the radius equals zero, the graph would be a point.

  16. An Application for Circles Seismologists can locate the epicenter of an earthquake by determining the intersection of three circles. The radii of these circles represent the distances from the epicenter to each of three receiving stations. The centers of the circles represent the receiving stations.

  17. Locating the Epicenter of an Earthquake Suppose receiving stations A, B, and C are located on a coordinate plane at the points (1, 4), (–3, –1,) and (5, 2). Let the distance from the earthquake epicenter to each station be 2 units, 5 units, and 4 units, respectively. Where on the coordinate plane is the epicenter located? Graph the three circles and determine the apparent epicenter.

  18. Locating the Epicenter of an Earthquake From graphing, it looks like the epicenter is at (1, 2). Check this algebraically. Station A: C: (1, 4), r = 2 Station B: C: (–3, –1), r = 5 Station C: C: (5, 2), r = 4 So, the point (1, 2) lies on all three graphs; we conclude that the epicenter is at (1, 2).

  19. Locating the Epicenter of an Earthquake • If three receiving stations at (1, 4), (–6, 0) and (5, –2) record distances to an earthquake epicenter of 4 units, 5 units, and 10 units, respectively, show algebraically that the epicenter lies at (–3, 4). Determine the equation for each circle and then substitute x = –3 and y = 4 to see if the point lies on all three graphs.

  20. Locating the Epicenter of an Earthquake • Station A: center (1, 4), radius 4 (–3, 4) lies on the circle.

  21. Locating the Epicenter of an Earthquake • Station B: center (–6, 0), radius 5 (–3, 4) lies on the circle.

  22. Locating the Epicenter of an Earthquake • Station C: center (5, –2), radius 10 (–3, 4) lies on the circle.

  23. Locating the Epicenter of an Earthquake • Since (–3, 4) lies on all three circles, it is the epicenter of the earthquake.

  24. An Application of Circles • When doing problems like this, show both the graph and the algebraic proof of your work.

  25. Assignment: Sec. 2.2: pp. 198-200 (13-16 all, 19-29 odd, 37-40 all)Due on Wednesday, 15 January 2014 • 13-16: Get from the book. • Write the equation. Then, decide whether the equation has a circle as its graph. If it does, give the center and radius. If it does not, describe the graph (a single point or non-existent). • 37-40: You do not have to write the problem, but you must graph and show the algebraic work. • 37) Suppose that receiving stations X, Y, and Z are located at the points (7, 4), (–9, –4), and (–3, 9), respectively. The epicenter of an earthquake is determined to be 5 units from X, 13 units from Y, and 10 units from Z. Where on the coordinate plane is the epicenter located?

  26. Assignment: Sec. 2.2: pp. 198-200 (13-16 all, 19-29 odd, 37-40 all)Due on Wednesday, 15 January 2014 • 38)Get from the book. • 39) The locations of three receiving stations and the distances to the epicenter of an earthquake are contained in the following three equations: (x – 2)2 + (y – 1)2 = 25,(x + 2)2 + (y – 2)2 = 16, and (x – 1)2 + (y + 2)2 = 9. Determine the location of the epicenter. • 40)Get from the book.