椭圆相关证明椭圆中任意一点M到右焦点的距离比M到右准线的距离为什么是a/c?

设：x^2/a^2+y^2/b^2=1
==>y^2=a^2-c^2-x^2+c^2x^2/a^2
M(x,y),右焦点F2(c,0),右准线:x=a^2/c
|MF2|^2=(x-c)^2+y^2
=x^2-2cx+c^2+a^2-c^2-x^2+c^2x^2/a^2
=a^2-2cx+c^2x^2/a^2
=c^2/a^2[a^4/c^2-2a^2x/c+x^2]
=c^2/a^2(a^2/c-x)^2
M到右准线的距离d^2=(a^2/c-x)^2
|MF2|^2/d^2 =c^2/a^2(a^2/c-x)^2/(a^2/c-x)^2
=c^2/a^2
==>|MF2|/d=c/a
椭圆中任意一点M到右焦点的距离比M到右准线的距离是c/a
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