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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