Suppose $x,y,z$ are real numbers satisfying, $$ x+y+z=0, \frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}=1 $$ for some numbers $0<a,b,c$ and $a+b+c=1$. I am trying to minimize the following quantity: $$ xyA+yzB+xzC, $$ for some numbers $A>0,B>0,C>0$.
When there is no constraint on the sum that $x+y+z=0$, I am able to deal it with matrix theory, that is, with some manipulation, I am going to find the minimum of eigenvalue of a particular matrix. However, adding this constraint, I have no good ways to solve it so far.
Thank you.