Hyperbola 1: Figure 12-4p shows the hyperbola 16x2 9y2 = 144. Its foci are at (5, 0) and

Chapter 12, Problem 1

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Hyperbola 1: Figure 12-4p shows the hyperbola 16x2 9y2 = 144. Its foci are at (5, 0) and (5, 0), and a directrix is the line x = 1.8. Its eccentricity is e = . Figure 12-4p a. A point on the hyperbola in Quadrant I has x-coordinate 7. Calculate y for this point. Does it agree with the graph? Store the answer as y in your grapher. b. Use the Pythagorean theorem and the result of part a to calculate these distances: d1 from the point (7, y) to the directrix d2 from the point (7, y) to the focus (5, 0) d3 from the point (7, y) to the focus (5, 0) c. Show that d2 = ed1. d. Show that |d2 d3| = 6, the length of the transverse axis (between the vertices). e. Find the x- and y-dilations. Which of these is the transverse radius, a, and which is the conjugate radius, b? f. As shown in Figure 12-4p, the focal radius is c = 5. Show that c2 = a2 + b2, the Pythagorean property for hyperbolas. g. Show that the directrix radius, d = 1.8, satisfies the equation a = ed and that the focal radius c = 5 satisfies c = ea.