Question

中心在原点,对称轴是坐标轴的双曲线与椭圆x^2/9+y^2/25=1的离心率之差为6/5,且过点A(3,4),球该双曲线

Answer 1

椭圆c1=√（25-9）=4,a1=5,b1=3,焦点在Y轴,

离心率e1=c1/a1=4/5,

双曲线离心率e2=e1+6/5=2,

若焦点在X轴,设双曲线方程为：x^2/a^2-y^2/b^2=1,

e2=c/a,c=2a,

b^2=c^2-a^2=4a^2-a^2=3a^2,

A坐标代入双曲线方程,

9/a^2-16/(3a^2)=1,

a^2=11/3,

c^2=4a^2=44/3,

b^2=c^2-a^2=11,

∴双曲线方程为：3x^2/11-y^2/11=1,

若焦点在Y轴,

设双曲线方程为：y^2/a^2-x^2/b^2=1,

e=c/a=2,

c=2a,

b^2=c^2-a^2=3a^2,

16/a^2-9/(3a^2)=1,

a^2=13,

c^2=4a^2=52,

b^2=c^2-a^2=39,

∴另一个双曲线方程为：y^2/13-x^2/39=1.