已知定点A(2,-1),B(-1,1),O为坐标原点,动点C满足向量OC=a向量OA+b向量OB,其中a,b属于R且2a

(1)
向量OC=x+yi
向量OA=2-i
向量OB=-1+i
x+yi=a(2-i)+b(-1+i)=(2a-b)+(b-a)i
x=2a-b
y=b-a
a=x+y
b=x+2y
(x+y)^2-(x+2y)^2=2
3y^2+2xy+2=0
(2)
3(kx-2)^2+2x(kx-2)+2=0
3k^2x^2-6kx+12+2kx^2-4x+2=0
(3k^2+2k)x^2-(6k+4)x+14=0
设M(x1,y1),N(x2,y2)
可解得
x1+x2=(6k+4)/(3k^2+2k)
x1x2=14/(3k^2+2k)
y1y2=k^2*x1x2-2k(x1+x2)+4
F(k)=|OM|*|ON|*cos∠MON
F(k)=(|OM|^2+|ON|^2-|MN|^2)/2
F(k)={x1^2+y1^2+x2^2+y2^2-[(x1-x2)^2+(y1-y2)^2]}/2
F(k)={x1^2+y1^2+x2^2+y2^2-[(x1-x2)^2+(y1-y2)^2]}/2
F(k)=(2x1x2+2y1y2)/2
F(k)=x1x2+y1y2
下面就不详细计算了