A Fairly Normal Asymmetry Problem

Calculus Level 4

Let \(P\) be a point (other than the origin) lying on the parabola \(y = x^{2}\). The normal line to the parabola at \(P\) will intersect the parabola at another point \(Q\). The minimum possible value for the area bounded by the line \(PQ\) and the parabola is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

Clarification: The normal line is the line perpendicular to the tangent line at a given point on a curve and which passes through the given point.

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