椭圆C:x²/a²+y²/2=1的焦点在x轴上，右顶点A为抛物线y²=16x的焦点（1）求椭圆C的方程
（2）已知点Q(-根号2,0），若斜率为根号-2/2的动直线l与椭圆C交与不同的两点M.N,求向量QM*向量QN的最小值。
抛物线焦点为（4,0），
∴a=4,椭圆方程为x^2/16+y^2/2=1。
(2)设l:y=-x/√2+m,
代入上式得x^2+8(x^2/2-√2mx+m^2)=16,
5x^2-8√2mx+8m^2-16=0,
设M(x1,y1),N(x2,y2),则
x1+x2=8√2m/5,x1x2=(8m^2-16)/5,
y1y2=(-x1/√2+m）（-x2/√2+m)
=x1x2/2-m(x1+x2)/√2+m^2,
向量QM*QN
=(x1+√2，y1)*(x2+√2，y2)
=(x1+√2）（x2+√2）+y1y2
=x1x2+√2（x1+x2)+2+x1x2/2-m(x1+x2...全部
抛物线焦点为（4,0），
∴a=4,椭圆方程为x^2/16+y^2/2=1。
(2)设l:y=-x/√2+m,
代入上式得x^2+8(x^2/2-√2mx+m^2)=16,
5x^2-8√2mx+8m^2-16=0,
设M(x1,y1),N(x2,y2),则
x1+x2=8√2m/5,x1x2=(8m^2-16)/5,
y1y2=(-x1/√2+m）（-x2/√2+m)
=x1x2/2-m(x1+x2)/√2+m^2,
向量QM*QN
=(x1+√2，y1)*(x2+√2，y2)
=(x1+√2）（x2+√2）+y1y2
=x1x2+√2（x1+x2)+2+x1x2/2-m(x1+x2)/√2+m^2
=3x1x2/2+(2-m√2）/2*（x1+x2)+2+m^2
=[3(8m^2-16)+8√2m(2-m√2）]/10+2+m^2
=[8m^2-16√2m-48]/10+2+m^2
=9m^2/5-8√2/5*m-14/5
=(9/5)(m-4√2/9）^2-158/45,
其最小值为-158/45。收起
