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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