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2014年全国高中数学联合竞赛(B卷)一试的最后一题是:如图,已知椭圆Γ:(x~2)/4+y~2=1,A(-2,0)、B(0,-1)是椭圆Γ上的两点,直线l_1:x=-2,l_2:y=-1。P(x_0,y_0)(x_0>0,y_0>0)是Γ上的一个动点,l_3是过点P且与Γ相切的直线,C、D、E分别是直线l_1与l_2、l_2与l_3、l_3与l_1的交点。求证:三条直线AD、BE、CP共点。这是一道解析几何综合题,涉及椭圆的切线、直线与直线、直线与椭圆的位置关系。人手并不难,但对运算和代数变形的要求较高。本文主要对该题的解法进行探究,并做适当的推广,供参考。
The last question of the 2014 National High School Mathematical Contest (B) is as follows: As shown, the ellipse Γ: (x ~ 2) / 4 + y ~ 2 = 1, A (-2,0), B (0, -1) is two points on ellipse Γ, straight line l_1: x = -2, l_2: y = -1. P (x_0, y_0) (x_0> 0, y_0> 0) is a moving point on Γ, l_3 is a straight line passing through point P and tangent to Γ, C, D and E are the straight lines l_1 and l_2, l_3, l_3 and l_1 of the intersection. Prove: three straight lines AD, BE, CP a total of points. This is a comprehensive analysis of geometric problems, involving the tangent of the ellipse, the line and the line, the relationship between the line and the ellipse. Manpower is not difficult, but the arithmetic and algebra deformation requirements are higher. This article mainly on the solution of the problem to explore, and make the appropriate promotion for reference.