Is the following set bounded?
$$\{(x,y): x^ 2 + xy + y^ 2 = 6\}$$
How can I draw it?
2026-04-04 14:45:35.1775313935
Is the following set bounded?
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Yes, it is bounded since it is an ellipse. If you make the substitution $x=u+v$, $y=u-v$, the equation becomes $$ (u+v)^2+(u+v)(u-v)+(u-v)^2=6, $$ which simplifies to $$ 3u^2+v^2=6, $$ or $$ \frac{u^2}{2}+\frac{v^2}6=1. $$ So your region is an ellipse, centered at the origin, with axes of length $2\sqrt2$ and $2\sqrt6$, rotated so that the axes are at 45 degrees from the $x$ and $y$ axes.