Question

请大家帮我解决一道抛物线的题,已知抛物线y^=2py（p>0）上两点A,B及顶点O满足∠AOB=90°,求弦AB的中点的

Answer 1

y²＝2px（p＞0)

设A（x1,y1）B(x2,y2).AB直线方程设为x=my+n带入抛物线方程得y²-2pmy-2pn=0.y1+y2=2pm

y1y2=-2pn.x1x2=(my1+n)(my2+n)=n².x1+x2=m(y1+y2)+2n=2pm²+2n.直线OA垂直OB,

得y1y2／x1x2=-1.即-2pn／（n²）=-1.得n=2p.AB中点坐标x=pm²+n=pm²+2p.y=pm.二者联立消去m得y²=px-2p²为所求轨迹方程.