Question

高中数学抛物线

线段AB是过抛物线X^2=2PY(P>O)焦点F的弦，M是抛物线的准线与Y轴的交点，O是坐标原点，过A,B两点分别作此抛物线的切线，两切线相交于N点。求证N点在抛物线的准线上

Answer 1

设A(x1,y1)B（x2,y2)
M(0,-p/2)
2x=2Py`
y`=x/P
y-y1=x1/P(x-x1)
y-y2=x2/P(x-x2)
x2y-x2y1=x1x2/P(x-x1)(1)
x1y-x1y2=x1x2/P(x-x2)(2)
(1)-(2)
(x2-x1)y+x1y2-x2y1=x1x2^2/P-x1^2x2/P
y2=x2^2/2P,y1=x1^2/2P
(x2-x1)y+x1x2^2/2P-x1^2x2/2P=x1x2^2/P-x1^2x2/P
(x2-x1)y=x1x2^2/2P-x1^2x2/2P=x1x2/2P(x2-x1)
x1/=x2
y=x1x2/2P
y-P/2=kx
x^2=2Py
x^2=2P(kx+P/2)
x^2=2pkx+P^2
x^2-2Pkx-P^2=0
x1+x2=2Pk
x1x2=-P^2
y=(-P^2)/2P=-P/2
证明完毕