SECTION 10.4 The Hyperbola 719 84. Sonic Boom Aircraft such as fighter jets routinely go supersonic (faster than the speed of sound). An aircraft moving faster than the speed of sound produces a coneshaped shock wave that “booms” as it trails the vehicle. The wave intersects the ground in the shape of one half of a hyperbola and the area over which the “boom” is audible is called the “boom carpet.” If an aircraft creates a shock wave that intersects the ground in the shape of the hyperbola x y 484 100 1 2 2 − = (units in miles), how wide is the “boom carpet” 32 miles behind the aircraft? 85. The eccentricity e of a hyperbola 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 c a, > it follows that e 1. > Describe the general shape of a hyperbola whose eccentricity is close to 1. What is the shape if e is very large? 86. A hyperbola for which a b = is called an equilateral hyperbola. Find the eccentricity e of an equilateral hyperbola. [Note:The eccentricity of a hyperbola is defined in Problem 85.] 87. Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas x y y x 4 1 and 4 1 2 2 2 2 − = − = are conjugate. Graph each hyperbola on the same set of coordinate axes. 88. Prove that the hyperbola y a x b 1 2 2 2 2 − = has the two oblique asymptotes y a b x y a b x and = = − 89. Challenge Problem Show that the graph of an equation of the form Ax Cy F A C F 0 0, 0, 0 2 2 + + = ≠ ≠ ≠ where A and C are opposite in sign, is a hyperbola with center at 0, 0 . ( ) 90. 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 opposite in sign, (a) is a hyperbola if D A E C F 4 4 0. 2 2 + − ≠ (b) is two intersecting lines if D A E C F 4 4 0. 2 2 + − = 81. Rutherford’s Experiment In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola. x 458 y (a) Find an equation of the asymptotes under this scenario. (b) If the vertex of the path of the alpha particles is 10 cm from the center of the hyperbola, find a model that describes the path of the particle. 82. Hyperbolic Mirrors Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically) pass through its rear focus, it is reflected through the front focus.This property, and that of the parabola, were used to develop the Cassegrain telescope in 1672. The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, then reflected toward the (common) focus, and thus are reflected by the hyperbolic mirror through the opening to its front focus, where the eyepiece is located. If the equation of the hyperbola is y x 9 16 1 2 2 − = and the focal length (distance from the vertex to the focus) of the parabola is 6, find the equation of the parabola. Source: www.enchantedlearning.com 83. Lamp Shadow The light from a lamp creates a shadow on a wall with a hyperbolic border. Find the equation of the border if the distance between the vertices is 18 inches and the foci are 4 inches from the vertices. Assume the center of the hyperbola is at the origin. Credit: cosma/Shutterstock
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