答:证明M点和Q点纵坐标相同即是命题.
① 求M点纵坐标:
设P点坐标是(t²/2p,t),t是参数.
PM斜率 k=t²/2pt = t/[t²/2p]=2p/t
PM直线方程:y = 2p/t x
准线x=-p/2
y = 2p/t * (-p/2) = -p²/t
② 求Q点纵坐标y`:
通过焦点的直线与抛物线交点y1和y2满足:
y1y2 = - p²
所以:
y`*t = -p²
y` = -p²/t
M点和Q点纵坐标相等.
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y1y2 = -p² 的证明:
y² = 2px
y = k(x - p/2)
(x - p/2)² = 2px/k²
x² -(p + 2p/k²)x + p²/4 = 0
x1x2 = p²/4
(2px1)(2px2)=p^4
y1²y2² =p^4
y1y2 = + - p² (y1、y2符号相反)
y1y2 = -p²
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