Question

已知双曲线x^2-(y^2)/2=1的焦点为F1、F2,点M在双曲线上且向量MF1点乘向量MF2＝0

Answer 1

C^2=a^2+b^2=1+2=3 c^2=3

向量MF1点乘向量MF2＝0,就是向量MF1点乘向量MF2垂直,

M点就是以F1,F2为直径的圆与x^2-y^2/2=1的交点:

圆心:(0,0) 半径平方=c^2=3

圆为x^2+y^2=3 x^2=3-y^2

与x^2-y^2/2=1的交点:

3-y^2-y^2/2=1

3/2y^2=2

y=2根号3/3 or y=-2根号3/3

x^2=3-4/3=5/3 x=-根号15/3 or x=根号15/3

M点坐标为:(,-根号15/3,2根号3/3) or ,-根号15/3,-2根号3/3)

(,根号15/3,2根号3/3) or (,-根号15/3,2根号3/3)