y^2=16x=2px,p=8,故焦点坐标是(4,0)
即有a=4,故椭圆方程是x^2/16+y^2/2=1.
(2)设直线L的方程是y=根号2/2X+b,代入到椭圆方程中得:
x^2/16+(x^2/2+根号2bx+b^2)/2=1
9x^2+16根号2bx+8b^2-16=0
x1+x2=-16根号2b/9,x1x2=(8b^2-16)/9
y1y2=(根号2/2X1+b)(根号2/2x2+b)=x1x2/2+根号2/2*(x1+x2)+b^2
向量QM=(x1+2根号2,y1)
向量QN=(x2+2根号2,y2)
QM*QN=(x1+2根号2)*(X2+2根号2)+y1y2
=x1x2+2根号2(x1+x2)+8+x1x2/2+根号2/2*(X1+X2)+b^2
=3/2*(8b^2--16)/9+5/2根号2*(-16根号2b/9)+8+b^2
=4/3b^2-8/3-80/9b+8+b^2
=7/3b^2-80/9b+16/3
=7/3(b-40/21)^2-592/189
当b=-(-80/9)/(2*7/3)=40/21时有最小值,是-592/189.