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众所周知,椭圆、双曲线、抛物线统称为圆锥曲线,它们有统一定义,且也有统一的极坐标方程,作为有心二次曲线的椭圆(包括圆)和双曲线,是否也有统一的定义、统一的方程呢? 设P_1、P_2是平面内的两定点,M为平面内的动点,有向直线P_1P_2到直线P_1M及直线P_2M的角分别为α_1,α_2,且tgα_1·tgα_2=k(k是非零常数)。动点M的轨迹是什么呢?
As we all know, ellipses, hyperbola, and parabola are collectively called conic curves. They have a unified definition, and they also have a unified polar coordinate equation. Whether or not ellipses (including circles) and hyperbolas that are concentric conics also have a unified definition and a unified equation Let P_1, P_2 be the two fixed points in the plane, and M be the moving point in the plane. The angles of the directed straight line P_1P_2 to the straight line P_1M and the straight line P_2M are α_1, α_2, respectively, and tgα_1·tgα_2 = k (k is a non-zero constant ). What is the trajectory of the moving point M?