过抛物线y∧2=2px(p>0)的焦点F的直线与抛物线相交于P,Q两点,线段PQ的中垂线交抛物线对称轴于R,求‖PQ‖=2‖FR‖
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![过抛物线y∧2=2px(p>0)的焦点F的直线与抛物线相交于P,Q两点,线段PQ的中垂线交抛物线对称轴于R,求‖PQ‖=2‖FR‖](/uploads/image/z/7863049-1-9.jpg?t=%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BFy%E2%88%A72%EF%BC%9D2px%28p%EF%BC%9E0%29%E7%9A%84%E7%84%A6%E7%82%B9F%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%B8%8E%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9B%B8%E4%BA%A4%E4%BA%8EP%2CQ%E4%B8%A4%E7%82%B9%2C%E7%BA%BF%E6%AE%B5PQ%E7%9A%84%E4%B8%AD%E5%9E%82%E7%BA%BF%E4%BA%A4%E6%8A%9B%E7%89%A9%E7%BA%BF%E5%AF%B9%E7%A7%B0%E8%BD%B4%E4%BA%8ER%2C%E6%B1%82%E2%80%96PQ%E2%80%96%EF%BC%9D2%E2%80%96FR%E2%80%96)
过抛物线y∧2=2px(p>0)的焦点F的直线与抛物线相交于P,Q两点,线段PQ的中垂线交抛物线对称轴于R,求‖PQ‖=2‖FR‖
过抛物线y∧2=2px(p>0)的焦点F的直线与抛物线相交于P,Q两点,线段PQ的中垂线交抛物线对称轴于R,求‖PQ‖=2‖FR‖
过抛物线y∧2=2px(p>0)的焦点F的直线与抛物线相交于P,Q两点,线段PQ的中垂线交抛物线对称轴于R,求‖PQ‖=2‖FR‖
设P点坐标(x1,y1)Q(x2,y2)
由抛物线且PQ过焦点F得‖PQ‖=‖PF‖+‖QF‖=x1+p/2+x2+p/2=x1+x2+p
PQ的斜率为k=(y2-y1)/(x2-x1)=(y2-y1)/[(y2^2-y1^2)/2p]=2p/(y1+y2)
∴PQ中垂线斜率为-1/k=-(y1+y2)/2p
PQ中垂线方程为y-(y1+y2)/2=-(y1+y2)/2p[x-(x1+x2)/2]
交对称轴x轴的交点坐标R为 令y=0 解得x=(x1+x2)/2+p
‖FR‖=(x1+x2)/2+p-p/2=(x1+x2+p)/2
故‖PQ‖=2‖FR‖