# Rolling Rugby

Calculus Level 3

As shown above is the elliptical graph of $$x^2 + xy + y^2 = 7$$, where $$P_{1}$$ and $$P_{2}$$ are the points on the graph with minimum and maximum $$x$$ coordinates respectively, and the red line $$l$$ is the linear graph passing through $$P_{1}$$ and $$P_{2}$$.

If the tangent parallel to line $$l$$ touches the graph at a point $$P_{3} (a,b)$$ for some real numbers $$a,b$$, compute $$a^2 + b^2$$.

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