1.
设A(x1,y1)、B(x2,y2)即A(x1,x12)、B(x2,x22)
△=k2+4c>0
x1+x2=k,x1·x2=-c,y1·y2=(x1·x2)2 =c2
x1x2+(x1·x2)2=c2-c=2
c=2,c=-1(舍去)
2.
过点C(0,2)直线L:y=kx+2 ...(1)
点A(a,a^2),B(b,b^2),点A在L上:a^2 =ka+2 ...(2)
(1)代入y=x^2:x^2-kx-2=0 ==> a+b=k
==> P(k/2,k^2/2 +2),Q(k/2,-2)
直线QA斜率 =(a^2+2)/(a-2) ...(3)
抛物线y=x^2在A(a,a^2)处切线斜率 =y'(a)=2a ...(4)
(2)(3)==> 直线QA斜率=2a =抛物线y=x^2在A(a,a^2)处切线斜率
==> QA为此抛物线的切线
3.
抛物线y=x^2在A(a,a^2)处切线方程：y=2ax-a^2 ...(5)
(5)与直线y=-2的交点:Q[(a^2-2)/(2a),-2]
过点Q、垂直于X轴的直线：x=(a^2-2)/(2a) ...(6)
直线AC：y=(a^2-2)x/a +2 ...(7)
(7)代入y=x^2:x^2-(a^2-2)x/a-2=0
==> 直线AC与抛物线两交点的中点的横坐标 =（a^2-2)/(2a)
中点在(6)上.证毕 逆命题成立