Directrix will Drive you Delirious

Algebra Level 5

The parabola \(P\) has equation \(y=x^2.\) The parabola \(P'\) is an image of \(P\) that was rotated clockwise about the focus of \(P\) a total of \(\theta\) degrees, where \(0 \le \theta \le 180^{\circ}.\) If \(P'\) is tangent to the directrix of \(P,\) then the smallest possible \(x\) value of all points on \(P'\) can be represented as \(-\dfrac{\sqrt{a}}{b}\) for positive integers \(a\) and \(b.\) Find \(a+b.\)

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