抛物线有关焦半径的结论

我只知道焦点弦的5条性质 y^2=2Px 过焦点F的直线交抛物线于A、B(1)|AB|=x1+x2+P=2P/sin^2(a)[a为直线AB的倾斜角](2)y1y2=-P^2 x1x2=P^2/4(3)1/|FA|+1/|FB|=2/P(4)以|AB|为直径的圆与抛物线的准线相切(5)焦半径公式：|AF|=x1+P/2证明：设A(x1,y1) B(x2,y2)(1)|AB|=|AF|+|BF|=x1-(-P/2)+x2-(-P/2)=x1+x2+P直线AB的斜率为tan(a) 则直线AB的方程为ycos(a)=sin(a)(x-P/2)与抛物线y^2=2Px联立 消去y 得sin^2(a)x^2-[2cos^2(a)+sin^2(a)]Px+P^2/4=0x1+x2=P[2cos^2(a)+sin^2(a)]/sin^2(a)∴|AB|=x1+x2+P={P[2cos^2(a)+sin^2(a)]+P[sin^2(a)]}/sin^2(a)=2P/sin^2(a)(2)设AB方程为x=ky+P/2 与抛物线方程y^2=2Px联立得y^2-2kPy-P^2=0 得y1y2=-P^2y1^2*y2^2=P^4=2Px1*2Px2=4P^2*x1x2得x1x2=P^2/4(3)1/|FA|+1/|FB|=1/(x1+P/2)+1/(x2+P/2)=(x1+x2+P)/(x1+P/2)(x2+P/2)=(x1+x2+P)/[x1x2+(x1+x2)P/2+p^2/4]=(x1+x2+P)/[P^2/2+(x1+x2)P/2]=2/P(4)圆心到准线的距离 d=(x1+x2)/2+P/2=|AB|/2(5)由抛物线定义直接得 补充：(1)设直线AB方程为y=k(x-P/2)1/kAM=(x1+P/2)/y1=(y1^2/2P+P/2)/y1=(y1^2+P^2)/2Py1同理可得 kBM=(y2^2+P^2)/2Py21/kAM+1/kBM=[(y1+y2)y1y2+(y1+y2)P^2]/2Py1y2∵y1+y2=2P/k y1y2=-P^2∴1/kAM+1/kBM=(y1+y2)(y1y2+P^2)/2Py1y2=0即kAM=-kBM 角AMF=角BMF(2)即证yN=yB 设A(y1^2/2P,y1)则直线OA可以写成 y=(2P/y1)*x 当x=-P/2时代人 得yN=-P^2/y1∵y1y2=-P^2∴yB=-P^2/y1=yN 得证