Question

已知双曲线与椭圆x^2/49+y^249=1共焦点,切以y=±4/3x为渐近线,求双曲线方程

Answer 1

x^2/49+y^2/24=1

则a^2=49,b^2=24,c^2=49-24=25

c=5

焦点坐标是:(-5,0),(5,0)

设双曲线方程是:x^2/a^2-y^2/b^2=1

渐近线方程是:y=(+/-)b/a*x=(+/-)4x/3.

故有:b/a=4/3,4a=3b

又c^2=a^2+b^2

25=a^2+(4a/3)^2

a^2=9

b^2=(4/3)^2a^2=16.

即方程是:x^2/9-y^2/16=1