Question

已知橢圓中心在原點,焦點在橫軸上,焦距為4,且和直線3x+2y-16=0相切,求橢圓方程.

Answer 1

橢圓的方程為+=1.

解析:

解法一:設橢圓方程為=1(a>b>0),

切點為P(x₀,y₀),則切線為=1.                                        ①

又切線為3x+2y-16=0,                                                     ②

故直線①②重合.

∴,即x₀=,y₀=b².

代入②,得9a²+28b²-256=0.                                                    ③

又焦距為4,∴c=2.

∴a²-b²=12.                                                                  ④

聯立方程③④,解得

故所求橢圓的方程為+=1.

解法二:c=2,c²=12.

設橢圓=1與直線方程聯立,Δ=0得b²=4.

故所求橢圓的方程為+=1.