最近笔者对椭圆、双曲线的焦半径和顶点弦垂直的关系作了些研究,得到一个十分有趣的充分必要条件.定理1 F是椭圆x2a2+2yb2=1(ab0)的焦点,P是椭圆上一点,A是椭圆长轴端点,且A与F位于y轴的两侧,直线PA的倾斜角为θ,椭圆的离心率为e,则PA·PF=0的充分必要条件是sinθ=1-ee.证明 Recently, the author has done some research on the relationship between the focal radius of the ellipse and the hyperbola and the perpendicularity of the vertex chord, and obtained a very necessary and sufficient condition. Theorem 1 F is the focus of the ellipse x2a2+2yb2=1(ab0), P is the ellipse One point, A is the endpoint of the major axis of the ellipse, and A and F are on both sides of the y-axis, the inclination angle of the straight line PA is θ, and the eccentricity of the ellipse is e. Then, the sufficient and necessary condition for PA·PF=0 is sin θ=1− Ee. Proof