# 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 $$a + b + c.$$

×