数学题——抛物线

(1)(y-y1)/(x-x1)=(y-y2)/(x-x2)
y1^2=2px1 y2^2=2px2 带入,得
y1/(p/2-y1^2/2p)=y2/(p/2-y2^2/2p)
化简,得 y1y2(y1-y2)=p^2(y2-y1)
y1y2不相等,则y1*y2=-p^2
带入抛物线方程中,得x1*x2=(p^2)/4
(2)直径d=根号下（x1-x2）^2+（y1-y2）^2
(x1+x2)^2-4x1x2+y1^2+y2^2-2y1y2
将(1)带入 (x1+x2)^2-p^2+2p(x1+x2)+2p^2
(x1+x2)^2+2p(x1+x2)+p^2
半径 r=(x1+x2+p)/2
圆心x=(x1+x2)/2
圆心与准线距离L=(x1+x2+p)/2=r
所以以AB为直径的圆必与抛物线的准线相切