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圆锥曲线是高考的重点和热点,圆锥曲线中蕴含着许多内涵丰富、结论优美的性质与结论,本文将介绍双曲线中与其渐近线有关的几个优美定值,以飨读者。结论 1:过双曲线x2a2-y2b2=1(a>0,b>0)上任意一点向两渐近线引垂线,垂足分别为M,N,则有如下结论:1PM·PN=a2·b2a2+b2;2P M·P N=a2b2(b2-a2)(a2+b2)2;3S△PMN=a3b3(a2+b2)2.证明:1设点P(x0,y0),所以b2x20-a2y20=a2b2,设半焦距为c,则a2+b2=c2,双曲线的渐近线方程为y=±b ax,所以过点P且与渐近线垂直的直线方程为y-y0=±a b,联立方程组可得M(a(ax0+by0)c2,b(ax0+by0)c2),N(a(ax0-by0)c2,-b(ax0-by0)c2).
Conic curve is the focus of college entrance examination and hot spots, the conic contains many connotations rich, concluding the nature and conclusions, this article will introduce hyperbolic several asymptotic lines related to its several beautiful settings to readers. Conclusion 1: Hyperbolism line at any point on the x2a2-y2b2 = 1 (a> 0, b> 0) leads the two asymptotes to the vertical and the foot-drop are respectively M and N, then the following conclusions: 1 PM · PN = a2 B2a2 + b2; 2PM.PN = a2b2 (b2-a2) (a2 + b2) 2; 3SΔPMN = a3b3 (a2 + b2) 2 Prove that 1 set point P (x0, y0), so b2x20- a2y20 = a2b2, assuming that the semi-focal distance is c, then a2 + b2 = c2, the asymptotic equation of the hyperbola is y = ± b ax, so the straight line equation of overexposed point P and perpendicular to the asymptotic line is y-y0 = ab, the system of simultaneous equations has M (ax0 + by0) c2, b (ax0 + by0) c2), N (a0-by0) c2, -b (ax0-by0) c2).