# Upstairs, downstairs .....

**Calculus**Level 5

Consider the parabola \(P\) with the equation \(x^{2} - 8x + 4y = 0.\) A line \(y = b\), with \(0 \lt b \lt 4\), is then drawn creating two regions \(R_{1}\) and \(R_{2}\) of equal area \(A\) where

\(R_{1}\) is the region bounded below by the line \(y = b\) and above by the curve \(P\), and

\(R_{2}\) is the region in the first quadrant bounded above by the line \(y = b\), to the left by the \(y\)-axis and to the right by the branch of \(P\) nearest the \(y\)-axis.

If \(A = \dfrac{a}{c}\), where \(a\) and \(c\) are positive coprime integers, then find \(a + b + c.\)

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