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 $a+b.$