732 CHAPTER 10 Analytic Geometry Analyzing and Graphing the Polar Equation of a Conic Analyze and graph the equation r 3 1 3cos . θ = + Solution EXAMPLE 3 The conic has its directrix perpendicular to the polar axis, a distance p units to the right of the pole. See Table 5. e ep p 3 and 3 1 = = = e 3 = Since e 3 1, = > the conic is a hyperbola with a focus at the pole. The directrix is perpendicular to the polar axis, 1 unit to the right of the pole.The transverse axis is along the polar axis.To find the vertices, let 0 θ = and . θ π = The vertices are 3 4 , 0 ( ) and 3 2 , . π ( ) − The center, which is at the midpoint of 3 4 , 0 ( ) and 3 2 , , π ( ) − is 9 8 , 0 . ( ) Then c distance = from the center to a focus 9 8 . = Using equation (2), we get a a 3 9 8 or 3 8 = = e c a = Then b c a b 81 64 9 64 72 64 9 8 3 2 2 3 2 4 2 2 2 = − = − = = = = Figure 60 shows the graph drawn by hand. Notice two additional points, 3, 2 π ( ) and 3, 3 2 , π ( ) are plotted on the left branch and symmetry is used to obtain the right branch. The asymptotes of the hyperbola were found by constructing the rectangle shown. Figures 61(a) and (b) show the graph of the equation using a TI-84 Plus CE in POLAR mode with min 0, max 2 , θ θ π = = and step 24 , θ π = using both dot mode and connected mode. Notice the extraneous asymptotes in connected mode. Figure 61(c) shows the graph using Desmos. Figure 60 θ = + r 3 1 3cos Polar axis O ( , 0) 3 – 4 ( , 0) 9 – 8 (2 , p) 3 – 2 (3, )p– 2 (3, ) 3p ––– 2 b 53 2 –––– 4 Now Work PROBLEM 17 Figure 61 r 3 1 3 cosθ = + 2 22 3.2 2 23.2 3.2 23.2 22 Connected mode (b) Desmos (c) (a) Dot mode 3 1 1 3cos u r 1 5 3 1 1 3cos u r 1 5
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