A circular window touches both the x-axis and the y-axis of a plan and lies in the first quadrant…

Step 1. The family. The centre is $(r, r)$ and the equation is $(x - r)^2 + (y - r)^2 = r^2$.

Step 2. The point. $(2 - r)^2 + (3 - r)^2 = r^2$ gives $4 - 4r + r^2 + 9 - 6r + r^2 = r^2$, so $r^2 - 10r + 13 = 0$.

Step 3. Solve. $r = \dfrac{10 \pm \sqrt{100 - 52}}{2} = 5 \pm 2\sqrt{3}$. So the radii are about $8.46$ and $1.54$ decimetres.

Step 4. The sum. The roots of $r^2 - 10r + 13 = 0$ add up to $10$.