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最近笔者对椭圆作了些研究 ,得到一个重要性质 ,现说明如下 .定理 设 L是过椭圆 x2a2 +y2b2 =1(a b 0 )的准线和 x轴交点 H的直线 ,L与椭圆相交于 A、B两点 ,O是坐标原点 ,c是半焦距 ,L的斜率为 k,则 OA .OB =0的充分必要条件是 k2 =b2 c2a4+b4.证明 不妨设 H为 (- a2c,0 )
Recently, the author has made some researches on ellipses and got an important property. Now it is explained as follows. Theorem Let L be the straight line of the crosshair x2a2 +y2b2 =1(ab 0) and the intersection point x of the x-axis, L intersects the ellipse in A. B, two points, O is the origin of the coordinates, c is the half focal length, and the slope of L is k, then OA .OB =0 is a necessary and sufficient condition that k2 = b2 c2a4+b4. The proof may be that H is (-a2c,0)