以椭圆x^/25+y^/9=1的长轴端点位焦点,且经过点p(5,9/4)求双曲线标准方程?

Question

以椭圆x^/25+y^/9=1的长轴端点位焦点,且经过点p(5,9/4)求双曲线标准方程?

Answer 1

x^2/25+y^2/9=1
a^2=25
a=5
双曲线的焦点：(-5,0)、(5,0)
a^2+b^2=25
b^2=25-a^2.(1)
x^2/a^2-y^2/b^2=1
过(5,9/4)
5^2/a^2-(9/4)^2/b^2=1
25/a^2-81/(16b^2)=1
25*16b^2-81a^2=16a^2b^2
400b^2-81a^2=16a^2b^2.(2)
(1)代入(2)：400(25-a^2)-81a^2=16a^2(25-a^2)
10000-400a^2-81a^2=400a^2-16a^4
16a^4-881a^2+10000=0
Δ=(-881)^2-4*16*10000=136161
√Δ=√136161=369
a^2=(881±369)/(2*32)
a^2=(881-369)/(2*32)=8
或者a^2=(881+369)/(2*32)=625/32
b^2=25-8=17
或者b^2=25-625/32=175/32
双曲线方程：x^2/8-y^2/17=1
或者x^2/(625/32)-y^2/(175/32)=1