<s>[INST] Give an upper bound for $x^2 + y^2$ for $(x, y)$ on the ellipse $(x - y - a)^2 + y^2 = 1$ without using Lagrange multipliers Let $a \in \mathbb{R}$ and consider the ellipse $(x - y - a)^2 + y^2 = 1$. My goal is to maximize (or at least give an upper bound for) $x^2 + y^2$ on this ellipse. This can be solved with Lagrange multipliers however the calculations are quite messy and hard to simplify (mostly due to $a$ being any real number). 
Are there any easier alternative ways that avoid using Lagrange multipliers, perhaps at the cost of showing instead there is a $C$ depending on $a$ such that $x^2 + y^2 \leq C$ for all $(x, y)$ on the ellipse?
 [/INST] This doesn't avoid Lagrange multipliers, but you can let $z=x-y$ and then 
try to find an upper bound for $(z+y)^2+y^2$ on the circle $(z-a)^2+y^2=1$:
1) If we maximize $(z+y)^2$ on the circle, we get 
$\hspace{.3 in}2(z+y)=\lambda\cdot2(z-a)$ and $2(z+y)=\lambda\cdot2y$, so the maximum occurs for $z=a+y$,
and then $(z-a)^2+y^2=1\implies y=\pm\frac{1}{\sqrt{2}}$ and $z=a\pm\frac{1}{\sqrt{2}}$.
Therefore $(z+y)^2+y^2\le (a+\sqrt{2})^2+y^2\le a^2+2\sqrt{2}a+2+1=\color{green}{a^2+2\sqrt{2}a+3}$
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