mx^2+ny^2=1
x+y=1
mx^2+n(1-x)^2=1
(m+n)x^2-2nx+n^2-1=0
x1+x2=2n/(m+n)
x1x2=(n^2-1)/(m+n)
(x1-x2)^2=(x1+x2)^2-4x1x2
=4n^2/(m+n)^2-4(n^2-1)/(m+n)
|AB|^2=2(x1-x2)^2=8
Cx=(x1+x2)/2=n/(m+n) Cy=1-n/(m+n) =m/(m+n)
Cy/Cx=m/n=√2/2
m=√2/2n
(x1-x2)^2=4 n^2/(m+n)^2-(n^2-1)/(m+n)=1
1/(1+√2/2)-(n^2-1)/(1+√2/2)n=1
n-(n^2-1)=(1+√2/2)n
1-n^2=√2n/2
(n+√2/4)^2=1+1/8
n+√2/4=3/√8
n=√2/2
m=1/2
x^2/2+y^2/√2=1