Geometricus Geometriorum (HP I, philosophical stone)

The ellipse equation shows that the minor radius and major radius are $\alpha = \sqrt{576} = 24$ and $\beta = \sqrt{625} = 25$ respectively.

The distance between a focus and the centre of the ellipse is given by $f = \sqrt{\beta^2 - \alpha^2} = \sqrt{625-576} = \sqrt{49} = 7$ , therefore, the distance between the two foci $a = 2 \times 7 = 14$ . ( See under Focus )

Let $x$ and $y$ be the coordinates of the given point $P$ , then we have:

$\begin{aligned} LHS & = \frac {\left(2017 + 24 \cos \frac \pi 4 - 2017\right)^2}{576} + \frac {\left(2016 + 25 \sin \frac \pi 4 - 2016\right)^2}{625} = \frac 12 + \frac 12 = 1 = RHS \end{aligned}$

This means that point $P$ is on the ellipse, and the sum of distances of any point on the ellipse to the foci is twice the major radius, that is $b = 2\beta = 2 \times 25 = 50$ . ( See under Focus )

$\implies a+b = 14+50 = \boxed{64}$ .