SECTION 1.6 Circles 55 63. Center ( ) −1, 3 and tangent to the line = y 2 64. Center ( ) − 4, 2 and tangent to the line = x 1 65. Challenge Problem If the equation of a circle is x y r 2 2 2 + = and the equation of a tangent line is = + y mx b, show that: (a) ( ) + = r m b 1 2 2 2 [ Hint : The quadratic equation ( ) + + = x mx b r 2 2 2 has exactly one solution.] (b) The point of tangency is ( ) −r m b r b , . 2 2 (c) The tangent line is perpendicular to the line containing the center of the circle and the point of tangency. 66. Challenge Problem Refer to Problem 65. The line − + = x y2 4 0 is tangent to a circle at ( ) 0, 2 . The line = − y x2 7 is tangent to the same circle at ( ) − 3, 1 . Find the center of the circle. 67. Challenge Problem The Greek Method The Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle, the line containing the center and the tangent line are perpendicular. Use this method to find an equation of the tangent line to the circle + = x y 9 2 2 at the point ( ) 1, 2 2 . 68. Challenge Problem Use the Greek method described in Problem 67 to find an equation of the tangent line to the circle + − + + = x y x y 4 6 4 0 2 2 at the point ( ) − 3, 2 2 3 . 69. Challenge Problem If ( ) x y , 1 1 and ( ) x y , 2 2 are the endpoints of a diameter of a circle, show that an equation of the circle is ( )( ) ( )( ) − − + − − = x x x x y y y y 0 1 2 1 2 70. Challenge Problem If + + + + = x y dx ey f 0 2 2 is the equation of a circle, show that ( ) ( ) + + + + + + = x x y y d x x e y y f 2 2 0 0 0 0 0 is an equation of the tangent line to the circle at the point ( ) x y , . 0 0 59. Vertically Circular Building Located in Al Raha,Abu Dhabi, the headquarters of property developing company Aldar is a vertically circular building with a diameter of 121 meters. The tip of the building is 110 meters aboveground. Find an equation for the building’s outline if the center of the building is on the y- axis. Credit: Pajor Pawel/Shutterstock 60. Vertically Circular Building The Sunrise Kempinski Hotel in Beijing, China, is a vertically circular building whose outline is described by the equation + − − = x y y 78 1843 0 2 2 if the center of the building is on the y -axis. If x and y are in meters, what is the height of the building? Problems 61–68 require the following discussion. The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency . See the figure. r y x In Problems 61–64, find the standard form of the equation of each circle. (Refer to the preceding discussion). 61. Center ( ) 2, 3 and tangent to the x -axis 62. Center ( ) −3, 1 and tangent to the y -axis ‘Are You Prepared?’ Answers 1. add; 25 2. { } −1, 5 73. Explain how the center and radius of a circle can be used to graph the circle. 74. What Went Wrong? A student stated that the center and radius of the graph whose equation is ( ) ( ) + + − = x y 3 2 16 2 2 are ( ) − 3, 2 and 4, respectively. Why is this incorrect? Explaining Concepts 71. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) ( ) ( ) − + + = x y 2 3 13 2 2 (b) ( ) ( ) − + − = x y 2 2 8 2 2 (c) ( ) ( ) − + − = x y 2 3 13 2 2 (d) ( ) ( ) + + − = x y 2 2 8 2 2 (e) + − − = x y x y 4 9 0 2 2 (f) + + − = x y x y 4 2 0 2 2 (g) + − − = x y x y 9 4 0 2 2 (h) + − − = x y x y 4 4 4 2 2 72. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) ( ) − + = x y 2 3 2 2 (b) ( ) + + = x y 2 3 2 2 (c) ( ) + − = x y 2 3 2 2 (d) ( ) + + = x y 2 4 2 2 (e) + + + = x y x 10 16 0 2 2 (f) + + − = x y x y 10 2 1 2 2 (g) + + + = x y x9 10 0 2 2 (h) + − − = x y x9 10 0 2 2 y x y x
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