Problem

The diagram shows a circle $C$ of radius $r$, where $x > 0$ and $y > 0$ for all points on $C$. The least distance between any point on $C$ and the $x$-axis is 8 units, and the least distance between any point on $C$ and the $y$-axis is 5 units.

(a) State the coordinates of the centre of the circle in terms of $r$.

(b) Given that the distance between the origin and the centre of the circle is 15 units, find the value of $r$.

(c) The point on the circle furthest from the origin is denoted by $P$.

Find the gradient of the tangent to the circle at $P$.