Question

若线段AB是抛物线y^2=2px的焦点弦,C为AB中点,C在准线的射影为C1,求证∠C1FC=90°

Answer 1

用向量解法.相当于求证：向量C1F⊥AB

设 A(x1,y1),B(x2,y2),准线为x=-p/2,焦点F(p/2,0)

则 C1(-p/2,(y1+y2)/2 )

所以 向量AB=(x2-x1,y2-y1),向量C1F=(p,-(y1+y2)/2 )

因为 AB•C1F=p(x2-x1) -(y2²-y1²)/2=p(x2-x1) -(2px2-2px1)/2=0

从而C1F⊥AB

所以∠C1FC=90°