# Problem of the Week Problem E and Solution Reach for the Sky

## Problem

The equation $$y=-5x^2+ax+b$$, where $$a$$ and $$b$$ are real numbers and $$a \ne b$$, represents a parabola. If this parabola passes through the points with coordinates $$(a,b)$$ and $$(b,a)$$, determine the maximum value of the parabola.

## Solution

Since $$(a,b)$$ lies on the parabola, it satisfies the equation of the parabola. We can substitute $$x=a$$ and $$y=b$$ into the equation $$y=-5x^2+ax+b$$. \begin{aligned} b&=-5a^2+a^2+b\\ b&=-4a^2+b\\ 0&=-4a^2\\ 0&=a^2\\ 0&=a\end{aligned} The equation becomes $$y=-5x^2+0x+b$$, or simply $$y=-5x^2+b$$.

Since $$(b,a)$$ lies on the parabola, it satisfies the equation of the parabola. We can substitute $$x=b$$ and $$y=a=0$$ into the equation $$y=-5x^2+b$$. \begin{aligned} 0&=-5b^2+b\\ 0&=b(-5b+1)\end{aligned} This means that $$b=0$$ or $$-5b+1=0$$. Therefore, $$b=0$$ or $$b=\frac{1}{5}$$.
Since $$a\ \ne b$$ and $$a=0$$, then $$b=0$$ is inadmissible.

Therefore, $$b=\frac{1}{5}$$ and the equation representing the parabola $$y=-5x^2+\frac{1}{5}$$. The parabola opens down and the vertex of the parabola is $$\left(0,\frac{1}{5}\right)$$, and so the maximum value of the parabola is $$\frac{1}{5}$$.