Question

设坐标原点是O,抛物线Y^2=2X与过焦点的直线交于AB两点,则向量OA乘以向量OB等于( ).

Answer 1

设直线l:y=k(x-1/2)

代入y^2=2x,得:k^2x^2-(k^2+2)x+k^2/4=0

设A(x1,y1),B(x2,y2)

则x1x2=1/4 x1+x2=(k^2+2)/k^2

y1y2=k^2(x1-1/2)(x2-1/2)=k^2[x1x2-1/2(x1+x2)+1/4]

=k^2[1/4-(k^2+2)/2k^2+1/4]

=1/2k^2(1-1-2/k^2)=-1

x1x2+y1y2=1/4-1=-3/4

当斜率不存在时,同理可得:x1x2+y1y2=-3/4

所以值为-3/4