26.We first prove a result which we will use: let m1and m2bethe slopes of two nonparallel, nonperpendicular lines. Let abe the acute angle between the lines. Then tan a5}1m112m1mm22}. (To see this result for positive-slope lines,let u1be the angle of inclination of the line with slope m1,and u2be the angle of inclination of the line with slope m2.Assume m1.m2. Then u1.u2and we have a5u12u2. Then tan a5tan (u12u2)55}1m112m1mm22},since m15tan u1and m25tan u2.)Now we prove the reflective property of ellipses (see theaccompanying figure):2b2x12a2yy9505⇒y952}ba22yx}. Let P(x0,y0) be any point on the ellipse ⇒y9(x0)52}aˇawb2xw2w0xw02w}52}ba22xy00}. Let F1(c, 0) andF2(2c, 0) be the foci. Then mPF15 }x0y20c}and mPF25 }x0y10c}. Let aand bbe the angles between thetangent line and PF1and PF2respectively. Then tan a555}cby20}.Similarly, tan b5 }cby20}. Since tan a5tan b, and aand bare both less than 908, we have a5b.27.To prove the reflective property for hyperbolas:b2x22a2y25a2b22b2x22a2yy0y95 }ba22xy}Let P(x0,y0) 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.