x^2/16+y^2/12=1
a^2=16,b^2=12,c=2
在l:X+Y-4=0上任意一点M
xM=n,yM=4-n
M(n,4-n)
过M(n,4-n)并且以椭圆x^2/16+y^2/12=1的焦点为焦点作椭圆
c=2,b^2=a^2-c^2=a^2-4>0
n^2/a^2+(4-n)^2/(a^2-4)=1
(a^2-4)*n^2+a^2*(4-n)^2=a^2*(a^2-4)
(2a^2-4)n^2-8a^2*n+20a^2-a^4=0
未知数为n的上方程有实数解,则它的判别式△≥0,即
(-8a^2)^2-4*(2a^2-4)*(20a^2-a^4)≥0
2a^6-28a^4+80a^2≥0
2a^2*(a^2-4)*(a^2-10)≥0
2a^2>0,b^2=a^2-4>0
∴a^2≥10
a^2最小=10,b^2=a^2-4=6
椭圆长轴最短a=√10
此椭圆方程:x^2/10+y^2/6=1
(2a^2-4)n^2-8a^2*n+20a^2-a^4=0
16n^2-80n+200-100=0
4n^2-20n+25=0
n=2.5,4-n=1.5
M(2.5,1.5)