Fun with conics

Geometry Level 5

Consider the parabola \(y^{2} = 4x\) and the ellipse \(\dfrac{x^{2}}{9} + \dfrac{y^{2}}{4} = 1\).

The tangent line of positive slope that is common to these two curves has a \(y\)-intercept of the form \(\sqrt{a + \sqrt{b}}\), where \(a\) and \(b\) are positive integers and \(b\) is square-free. Find \(a + b\).

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