Rolling Rugby

Calculus Level 2

As shown above is the elliptical graph of x2+xy+y2=7x^2 + xy + y^2 = 7, where P1P_{1} and P2P_{2} are the points on the graph with minimum and maximum xx coordinates respectively, and the red line ll is the linear graph passing through P1P_{1} and P2P_{2}.

If the tangent parallel to line ll touches the graph at a point P3(a,b)P_{3} (a,b) for some real numbers a,ba,b, compute a2+b2a^2 + b^2.

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