过点（0,-1/2）的直线l与抛物线 ：y=-x*x交于a,b两点,o为坐标原点,则oa的向量乘以ob的值为?

题目

过点（0,-1/2）的直线l与抛物线 ：y=-x*x交于a,b两点,o为坐标原点,则oa的向量乘以ob的值为?

答案

P: y = -x^2 (1)
(0,-1/2) pass through L
P and L intersact at A, B
O is origin
To find : OA . OB
let L be
y = mx +c
(0,-1/2)
c = -1/2
=> L: y= mx-1/2 (2)
Sub (2) into (1)
mx-1/2 = -x^2
x^2+mx - 1/2 =0
let A(x1,y1), B(x2,y2)
x1+x2 = -m
x1x2 = -1/2
Similarly we have
y = -(y+1/2)^2/m^2
y^2+ (m^2+1)y + 1/4 =0
y1+y2 = -(m^2+1)
y1y2= 1/4
OA.OB
=(x1,y1).(x2,y2)
=x1x2 + y1y2
= -1/2 + 1/4
=-1/4