Question

在双曲线x²-y²=1上求一点P,使它与该双曲线的两焦点F1,F2的连线互相垂直

Answer 1

设P(x,y) F1(-√2,0) F2(√2,0)

kPF1=y/(x-√2)

kPF2=y/(x+√2)

kPF1*kPF2=y^2/(x^2-2)=-1

y^2+x^2=2

x^2-y^2=1

2x^2=3

x1=-√6/2 y=√2/2或y=-√2/2

x2=√6/2 y=√2/2或y=-√2/2

所以P点有四个,坐标为(±√6/2,±√2/2)