704 CHAPTER 10 Analytic Geometry 91. Challenge Problem Show that an equation of the form Ax Cy F A C F 0 0, 0, 0 2 2 + + = ≠ ≠ ≠ where A and C are of the same sign and F is of opposite sign, (a) is the equation of an ellipse with center at 0, 0 ( ) if A C. ≠ (b) is the equation of a circle with center 0, 0 ( ) if A C. = 92. Challenge Problem Show that the graph of an equation of the form Ax Cy Dx Ey F A C 0 0, 0 2 2 + + + + = ≠ ≠ where A and C are of the same sign, (a) is an ellipse if D A E C F 4 4 2 2 + − is the same sign as A. (b) is a point if D A E C F 4 4 0. 2 2 + − = (c) contains no points if D A E C F 4 4 2 2 + − is of opposite sign to A. If the equation of the ellipse formed by the reflector is x y 324 100 1, 2 2 + = how far from the kidney stone does the shock wave generator need to be placed? (Units are in centimeters.) 88. Elliptical Trainer The pedals of an elliptical exercise machine travel an elliptical path as the user is exercising. If the stride length (length of the major axis) for one machine is 20 inches and the maximum vertical pedal displacement (length of the minor axis) is 9 inches, find the equation of the pedal path, assuming it is centered at the origin. 89. Challenge Problem For the ellipse, x y5 20, 2 2 + = let V be the vertex with the smaller x-coordinate and let B be the endpoint on the minor axis with the larger y-coordinate. Find the y-coordinate of the point M that is on the line x 5 0 + = and is equidistant from V and B. 90. Challenge Problem Consider the circle x y 2 1 2 2 ( ) − + = and the ellipse with vertices at 2, 0 ( ) and 6, 0 ( ) and one focus at 4 3, 0 . ( ) + Find the points of intersection of the circle and the ellipse.† Explaining Concepts 93. The eccentricity e of an ellipse is defined as the number c a , where a is the distance of a vertex from the center and c is the distance of a focus from the center. Because a c, > it follows that e 1. < Write a brief paragraph about the general shape of each of the following ellipses. Be sure to justify your conclusions. (a) Eccentricity close to 0 (b) Eccentricity 0.5 = (c) Eccentricity close to 1 †Courtesy of the Joliet Junior College Mathematics Department Retain Your Knowledge Problems 94–103 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. ‘Are You Prepared?’ Answers 1. 13 2. 9 4 3. 2,0, 2,0, 0, 4, 0,4 ( ) ( ) ( ) ( ) − − 4. 2, 5 ( ) 5. left; 1; down; 4 6. x y 2 3 1 2 2 ( ) ( ) − + + = 94. Find the zeros of the quadratic function f x x 5 12. 2 ( ) ( ) = − − What are the x-intercepts, if any, of the graph of the function? 95. Find the domain of the rational function f x x x 2 3 5 . ( ) = − − Find any horizontal, vertical, or oblique asymptotes. 96. Find the work done by a force of 80 pounds acting in the direction of 50° to the horizontal in moving an object 12 feet from 0, 0 ( ) to 12, 0 . ( ) Round to one decimal place. 97. Solve the right triangle shown. 14 528 b B c 98. Solve ( ) + = x 2 3 tan 5 7 9 for x 0 2 . π ≤ < 99. What value does R x x x x x 3 14 8 12 2 2 ( ) = + + + − approach as x 4? →− 100. Solve e 8 x2 1 = − + rounded to four decimal places. 101. Find the difference quotient of f x x x h 2 7 as 0. 2 ( ) = − → 102. Solve: x log 2 1 4 3( ) − = 103. Solve: x 3 20 2 ( ) + =
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