Question

已知双曲线焦点在x轴上,且过点(－√2,－√3),（1,1）,求双曲线c的标准方程.怎么解,

Answer 1

据题意,设双曲线方程：x^2/a^2-y^2/b^2=1

b^2x^2-a^2y^2=a^2b^2

过(-√2,-√3)

b^2(-√2)^2-a^2(-√3)^2=a^2b^2

2b^2-3a^2=a^2b^2

(2-a^2)b^2=3a^2

b^2=3a^2/(2-a^2).(1)

过(1,1)

b^2*1^2-a^2*1^2=a^2b^2

b^2-a^2=a^2b^2.(2)

(1)代入(2)：3a^2/(2-a^2)-a^2=a^2*3a^2/(2-a^2)

3a^2-a^2(2-a^2)=3a^4

3a^2-2a^2+a^4=3a^4

a^2=2a^4

1=2a^2

a^2=1/2

代入(1)：b^2=(3*1/2)/(2-1/2)=1

双曲线方程：x^2/(1/2)-y^2=1