Interesting Observation

That certainly is interesting! Here's a sketch of one approach:

• rearrange the expression for $y$ to find $\cos{t}=\frac{2y+15}{3+y}$

• substitute this into the expression for $x$ , and rearrange to find $\sin{t}=x-\frac{(2y+15)x}{2(3+y)}+1$

• square these two equations and add, then tidy up: $81x^2-36xy+16y^2-108x+240y+900=0$

which is indeed the equation of an ellipse, in a slightly more familiar guise. We then work out the axes from here - using, for example, this . The two axes are

$\frac{\sqrt{2(97\pm \sqrt{5521})}}{6}$

with numerical values $3.0849\ldots$ and $1.1229\ldots$ leading to the answer $\boxed{4207}$ .

I strongly suspect there's a better approach than this!