Pre-Calc Homework Solutions 463

Pre-Calc Homework Solutions 463 - Appendix A5.2 26 We first...

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26. We first prove a result which we will use: let m 1 and m 2 be the slopes of two nonparallel, nonperpendicular lines. Let a be the acute angle between the lines. Then tan a 5 } 1 m 1 1 2 m 1 m m 2 2 } . (To see this result for positive-slope lines, let u 1 be the angle of inclination of the line with slope m 1 , and u 2 be the angle of inclination of the line with slope m 2 . Assume m 1 . m 2 . Then u 1 . u 2 and we have a 5 u 1 2 u 2 . Then tan a 5 tan ( u 1 2 u 2 ) 55 } 1 m 1 1 2 m 1 m m 2 2 } , since m 1 5 tan u 1 and m 2 5 tan u 2 .) Now we prove the reflective property of ellipses (see the accompanying figure): 2 b 2 x 1 2 a 2 yy 95 0 5 y 952} b a 2 2 y x } . Let P ( x 0 , y 0 ) be any point on the ellipse y 9 ( x 0 ) 52 } a ˇ a w b 2 x w 2 w 0 x w 0 2 w }52} b a 2 2 x y 0 0 } . Let F 1 ( c , 0) and F 2 ( 2 c , 0) be the foci. Then m PF 1 5 } x 0 y 2 0 c } and m PF 2 5 } x 0 y 1 0 c } . Let a and b be the angles between the tangent line and PF 1 and PF 2 respectively. Then tan a 5 5 5 } c b y 2 0 } . Similarly, tan b 5 } c b y 2 0 } . Since tan a 5 tan b , and a and b are both less than 90 8 , we have a 5 b . 27. To prove the reflective property for hyperbolas: b 2 x 2 2 a 2 y 2 5 a 2 b 2 2 b 2 x 2 2 a 2 yy 0 y 95 } b a 2 2 x y } Let P ( x 0 , y 0 ) be a point of tangency (see the accompanying
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.

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