Question

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?

Answer 1

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