1.
設(shè)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.
過點C(0,2)直線L:y=kx+2 ...(1)
點A(a,a^2),B(b,b^2),點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)
直線QA斜率 =(a^2+2)/(a-2) ...(3)
拋物線y=x^2在A(a,a^2)處切線斜率 =y'(a)=2a ...(4)
(2)(3)==> 直線QA斜率=2a =拋物線y=x^2在A(a,a^2)處切線斜率
==> QA為此拋物線的切線
3.
拋物線y=x^2在A(a,a^2)處切線方程：y=2ax-a^2 ...(5)
(5)與直線y=-2的交點:Q[(a^2-2)/(2a),-2]
過點Q、垂直于X軸的直線：x=(a^2-2)/(2a) ...(6)
直線AC：y=(a^2-2)x/a +2 ...(7)
(7)代入y=x^2:x^2-(a^2-2)x/a-2=0
==> 直線AC與拋物線兩交點的中點的橫坐標 =（a^2-2)/(2a)
中點在(6)上.證畢 逆命題成立