When doing homework for convex optimization I got stuck in the middle of my work with this problem: I need to find the vertices of this ellipse to continue: $$25x^2+48xy+25y^2=1225$$
Does anyone know how to find them? Thx a lot
2026-04-25 14:03:37.1777125817
Finding vertex of ellipse given its non-trivial equation
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Make the change of variables: $u=x+y$ and $v=x-y$. The equation of the ellipse is then $$ {25\over2}(u^2+v^2)+12(u^2-v^2)=1225, \quad\hbox{that is}\quad 49u^2+v^2=2450, $$ whose vertices are readily found at $u=\pm\sqrt{2450/49}=\pm\sqrt{50}$ and $v=\pm\sqrt{2450}=\pm7\sqrt{50}$. Change back to $x$, $y$ and you're done.