HMMT 2015 G10

Let $\mathcal{G}$ be the set of all points $(x, y)$ in the Cartesian plane such that $0 \leq y \leq 8$ and

$$(x-3)^{2}+31=(y-4)^{2}+8 \sqrt{y(8-y)}$$

There exists a unique line $\ell$ of negative slope tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a unique point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.