过椭圆（x^2/a^2+y^2/b^2=1）右焦点向它的动切线引垂线 垂足方程

垂足(x,y),c>0,右焦点(c,0),设a>b,则c=√(a^2-b^2)
设动切线垂线:y=kx+m
椭圆x^2/a^2+y^2/b^2=1
b^2*x^2+a^2*y^2=(ab)^2
b^2*x^2+a^2*(kx+m)^2=(ab)^2
(b^2+a^2*k^2)*x^2+2a^2*kmx+(am)^2-(ab)^2=0
y=kx+m与椭圆（x^2/a^2+y^2/b^2=1）相切，上方程的判别式△=0，即
(2a^2*km)^2-4(b^2+a^2*k^2)*[(am)^2-(ab)^2]=0
m^2=b^2+(ak)^2
m=±√[b^2+(ak)^2]
y=kx±√[b^2+(ak)^2]
y-kx=±√[b^2+(ak)^2]
y^2+x^2*k^2-2xyk=b^2+a^2*k^2
(x^2-a^2)k^2-2xyk+y^2-b^2=0
b^2*x^2+a^2*y^2=(ab)^2
动切线k1=[xy±√(x^2*b^2+a^2*y^2-a^2*b^2)]/(x^2-a^2)=xy/(x^2-a^2)
动切线垂线k2=y/(x-c)
k2*k1=-1
[y/(x-c)]*[xy/(x^2-a^2)]=-1
x^3+(a^2-x^2)c+xy^2-xa^2=0
x^3+(a^2-x^2)*√(a^2-b^2)+xy^2-xa^2=0
方法正确，请检验一下。