Question

参数方程设p是双曲线bx^2-a^2y^2=a^2b^2（a>0 b>0）上任意一点,过p做双曲线两条渐近线的平行线,—

Answer 1

设P(X,Y),则 (bX)^2-(aY)^2=(ab)^2.

两条渐近线,y=bx/a,y=-bx/a;

PQ,y-Y=b(x-X)/a; PR,y-Y=-b(x-X)/a;

{y-Y=b(x-X)/a,y=-bx/a} ==> Q((aY+bX)/(2b),(aY+bX)/(2a)).

{y-Y=-b(x-X)/a,y=bx/a} ==> R((-aY+bX)/(2b),(aY-bX)/(2a)).

(PQ*PR)^2=[((aY+bX)/(2b)-X)^2+((aY+bX)/(2a)-Y)^2]*[((-aY+bX)/(2b)-X)^2+((aY-bX)/(2a)-Y)^2]

=[(-aY+bX)^2*(a^2+b^2)/(2ab)^2]*[(aY+bX)^2*(a^2+b^2)/(2ab)^2]

=[(bX)^2-(aY)^2]*(a^2+b^2)^2/(2ab)^4

=(a^2+b^2)^2/(4ab)^2,

==> PQ*PR=(a^2+b^2)/(4ab).

(ps,题目错了吧!)