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thomasET_226348_ism48

thomasET_226348_ism48 - 640 Chapter 10 Conic Sections and...

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640 Chapter 10 Conic Sections and Polar Coordinates 14. P traces a hypocycloid where the larger radius is 2a and the smaller is a x (2a a) cos a cos Ê œ ) ) ˆ 2a a a 2a cos , 0 2\ , and y (2a a) sin a sin a sin a sin 0. Therefore P traces the œ Ÿ Ÿ œ œ œ ) ) 1 ) ) ) ) ˆ 2a a a diameter of the circle back and forth as goes from 0 to 2 . ) 1 15. Draw line AM in the figure and note that AMO is a right n angle since it is an inscribed angle which spans the diameter of a circle. Then AN MN AM . Now, OA a, # # # œ œ tan t, and sin t. Next MN OP AN AM a a œ œ œ OP AN AM a tan t a sin t Ê œ œ # # # # # # # OP a tan t a sin t Ê œ È # # # # (a sin t) sec t 1 . In triangle BPO, œ œ È # a sin t cos t x OP sin t a sin t tan t and œ œ œ a sin t cos t # y OP cos t a sin t x a sin t tan t and y a sin t. œ œ Ê œ œ # # # 16. Let the x-axis be the line the wheel rolls along with the y-axis through a low point of the trochoid (see the accompanying figure). Let denote the angle through which the wheel turns. Then h a and k a. Next introduce x y -axes ) ) œ œ w w parallel to the xy-axes and having their origin at the center C of the wheel.
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