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 Êœ ± ² )) 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 ) ) ) ) 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, ### œ² œ tan t, and sin t. Next MN OP AN AM aa œœ œ OP AN AM a tan t a sin t Êœ±œ ± # # # # OP a tan t a s in t Êœ ± 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 a sin t cos t $ # y OP cos t a sin t x a sin t tan t and y a sin t. 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).
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This note was uploaded on 04/17/2008 for the course MA 113 taught by Professor Massman during the Spring '08 term at Rose-Hulman.

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thomasET_226348_ism48 - 640 Chapter 10 Conic Sections and...

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