y^2=16x=2px,p=8,故焦點(diǎn)坐標(biāo)是(4,0)
即有a=4,故橢圓方程是x^2/16+y^2/2=1.
(2)設(shè)直線L的方程是y=根號(hào)2/2X+b,代入到橢圓方程中得:
x^2/16+(x^2/2+根號(hào)2bx+b^2)/2=1
9x^2+16根號(hào)2bx+8b^2-16=0
x1+x2=-16根號(hào)2b/9,x1x2=(8b^2-16)/9
y1y2=(根號(hào)2/2X1+b)(根號(hào)2/2x2+b)=x1x2/2+根號(hào)2/2*(x1+x2)+b^2
向量QM=(x1+2根號(hào)2,y1)
向量QN=(x2+2根號(hào)2,y2)
QM*QN=(x1+2根號(hào)2)*(X2+2根號(hào)2)+y1y2
=x1x2+2根號(hào)2(x1+x2)+8+x1x2/2+根號(hào)2/2*(X1+X2)+b^2
=3/2*(8b^2--16)/9+5/2根號(hào)2*(-16根號(hào)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
當(dāng)b=-(-80/9)/(2*7/3)=40/21時(shí)有最小值,是-592/189.