Determine the largest possible radius of a circle that is tangent to both the $x$-axis and $y$-axis, and passes through the point $(9,2)$.

(A circle in the $x y$-plane with centre $(a, b)$ and radius $r$ has equation $(x-a)^{2}+(y-b)^{2}=r^{2}$.)
If a circle is tangent to both the $x$-axis and the $y$-axis, it must be entirely contained in one of the quadrants. Since we are looking for circles that pass through $(9,2)$, they must be entirely in the first quadrant.

Consider a circle of radius $r$ that is tangent to both the $x$-axis and the $y$-axis and is in the first quadrant.

Since the circle is tangent to the $x$-axis, its centre must be $r$ units from the $x$-axis. Similarly, its centre must be $r$ units from the $y$-axis, so the coordinates of the centre must be $(r, r)$.

A circle with centre $(r, r)$ and radius $r$ has equation $(x-r)^{2}+(y-r)^{2}=r^{2}$.

For this circle to pass through $(9,2)$, the pair $(9,2)$ must satisfy the equation of the circle, so

$$
\begin{aligned}
(9-r)^{2}+(2-r)^{2} & =r^{2} \\
r^{2}-18 r+81+r^{2}-4 r+4 & =r^{2} \\
r^{2}-22 r+85 & =0 \\
(r-17)(r-5) & =0
\end{aligned}
$$

Therefore, $r=5$ or $r=17$, and so the largest possible radius is 17 .

ANSWER: 17