x^2/4+y^2/3=1
1参数方程 a=2,b=√3
x=2cosu
y=√3sinu
x+2y=2cosu+2√3sinu=4sin(u+π/6)
-1≤sin(u+π/6)≤1
-4≤x+2y≤4
2
MN直线方程： y=k(x-2√3) k=tana M(x1,y1) N(x2,y2)
3x^2+4y^2-12=0
3x^2+4k^2(x-2√3)^2-12=0
(3+4k^2)x^2-16√3k^2x+48k^2-12=0
x1+x2=16√3k^2/(3+4k^2)
x1x2=(48k^2-12)/(3+4k^2)
|QM|^2=(x1-2√3)^2+y1^2=(x1-2√3)^2+k^2(x1-2√3)^2=(k^2+1)(x1-2√3)^2
|QN|^2=(x2-2√3)^2+y2^2=(x2-2√3)^2+k^2(x2-2√3)^2=(k^2+1)(x2-2√3)^2
|QM||QN|=(k^2+1)|(x1-2√3)(x2-2√3)|=(k^2+1)|x1x2-2√3(x1+x2)+12|
=(k^2+1)|[(48k^2-12)-96k^2]/(3+4k^2) +12|
=(k^2+1)*24/(3+4k^2)
=24*(k^2+1)/(3+4k^2)
k^2+1=(tana)^2+1=1/(cosa)^2
3+4k^2=3+4(tana)^2=4+4(tana)^2-1=4/(cosa)^2-1
(k^2+1)/(3+4k^2)=1/[4-(cosa)^2]
|QM|*|QN|=24*(k^2+1)/(3+4k^2)=24/[4-(cosa)^2]
0≤(cosa)^2≤1
24/5≤|QM|*|QN|≤6