# Minimizing an Area

Algebra Level 1

Consider two points $$A=(4, -4)$$ and $$B=(8, 0),$$ and a third point $$P$$ on the graph of $$y=\frac{1}{2}x^2$$. If $$P=(a,b)$$ is the point minimizing the area of the triangle $$ABP,$$ and $$S$$ is the minimum area, what is the value of $$a+2b+4S?$$

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