请教一个椭圆中点弦轨迹的问题,

Question

请教一个椭圆中点弦轨迹的问题,

Answer 1

设M(x,y),P(x+p,y+q),Q(x-p,y-q), 由P,Q在椭圆x^2/3+y^2=1上,得 （x+p)^2/3+(y+q)^2=1, (x-p)^2/3+(y-q)^1=1, 相加,除以2,得（x^2+p^2)/3+y^2+q^2=1,① 相减得4px/3+4qy=0,p/q=-3y/x,② 由OP⊥OQ得 （x+p)(x-p)+(y+q)(y-q)=0, x^2+y^2-p^2-q^2=0,③ ①+③,4x^2/3+2y^2-2p^2/3=1, ∴p^2=2x^2+3y^2-3/2, 代入③,q^2=3/2-x^2-2y^2, 由②,（2x^2+3y^2-3/2)/(3/2-x^2-2y^2)=9y^2/x^2, ∴4x^4+6x^2y^2-3x^2=27y^2-18x^2y^2-36y^4, 即4x^4+24x^2y^2+36y^4-3x^2-27y^2=0,为所求.