42. A circle with center C(-4, 0) has radius 15. a. Verify that A(8,9) and B(-13,...

  

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42. A circle with center C(-4, 0) has radius 15. a. Verify that A(8,9) and B(-13, 12) are points on this circle. b. Write an equation of the perpendicular bisector of AB and show that the coordinates of point C satisfy the equation. c. What theorem from geometry does this exercise illustrate? 43. A diameter of a circle has endpoints A (13, 0) and B(-13,0). a. Show that P(-5, 12) is a point on this circle. b. Show that PA and PB are perpendicular. 44. a. Find the coordinates of A and of B if AB is a horizontal diameter of the circle x + y - 34x = 0. b. Show that P(2, 8) is a point on this circle and that PAL PB. 45. Given 0(0, 0) and N(12, 0), find an equation in terms of x and y for all points P(x, y) such that PO 1 PN. Simplify this equation and show that P is on a circle. What are the center and radius of the circle? 46. Discussion Given A(6, 8) and B(-6, -8), write an equation in terms of x and y for all points P(x, y) such that PA PB. Simplify the equation and interpret your answer. You may find it helpful to have a graphing calculator to complete 47. A triangle is inscribed in a semicircle as shown at the right. a. Find an equation for the semicircle. b. Write a function A(x) for the area of the triangle. c. What is the domain of the function from pat (b)? d. Graph-y= A(x). e. Use the graph from part (d) to find the value of x that maximizes A(x). P(X.31 42. A circle with center C(-4, 0) has radius 15. a. Verify that A(8,9) and B(-13, 12) are points on this circle. b. Write an equation of the perpendicular bisector of AB and show that the coordinates of point C satisfy the equation. c. What theorem from geometry does this exercise illustrate? 43. A diameter of a circle has endpoints A (13, 0) and B(-13,0). a. Show that P(-5, 12) is a point on this circle. b. Show that PA and PB are perpendicular. 44. a. Find the coordinates of A and of B if AB is a horizontal diameter of the circle x + y - 34x = 0. b. Show that P(2, 8) is a point on this circle and that PAL PB. 45. Given 0(0, 0) and N(12, 0), find an equation in terms of x and y for all points P(x, y) such that PO 1 PN. Simplify this equation and show that P is on a circle. What are the center and radius of the circle? 46. Discussion Given A(6, 8) and B(-6, -8), write an equation in terms of x and y for all points P(x, y) such that PA PB. Simplify the equation and interpret your answer. You may find it helpful to have a graphing calculator to complete 47. A triangle is inscribed in a semicircle as shown at the right. a. Find an equation for the semicircle. b. Write a function A(x) for the area of the triangle. c. What is the domain of the function from pat (b)? d. Graph-y= A(x). e. Use the graph from part (d) to find the value of x that maximizes A(x). P(X.31