# The not-so-random tangent line meeting.

Level pending

We are given the curve (parabola) $$y=ax^{2}$$, where $$a\neq 0$$. Let us take $$2$$ random tangent lines to the given parabola, $$t_{1}$$ and $$t_{2}$$, that have their points of tangency at the points $$A$$ and $$B$$. These two tangent lines intersect at a certain point that we will call $$C$$. The vertical line that goes through $$C$$ cuts the line segment $$AB$$ in the point $$C'$$. If the value of $$\frac{\overline{AC'}}{\overline{AB}}$$ is $$\frac{a}{b}$$, where $$a$$ and $$b$$ are two co-prime positive integers, what is the value of $$a+b$$?

Details and assumptions:

A point of tangency is the point at which a particular tangent line intersects its curve.

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