# Circling a parabola!

Geometry Level 4

From the vertex $$O$$ of the parabola $$y^2 = 4ax$$, two distinct chords are drawn which intersect the parabola at two distinct points $$P$$ and $$Q$$. Two circles are then drawn with chords $$OP$$ and $$OQ$$ as their respective diameters. These circles intersect each other at two points, at $$O$$ and at another point inside the parabola, $$R$$. Let $$\theta_1$$ and $$\theta_2$$ be the respective gradients of the tangents to the parabola at $$P$$ and $$Q$$ respectively and $$\varphi$$ be the gradient of the line $$OR$$. Find the value of $$\cot \theta_1 + \cot \theta_2$$ in terms of $$\varphi$$.

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