## Coordinate of a Point on a Circle Furthest From the Origin

For the circle

\[(x-2)^2+(y-3)^2=5^2\]

with centre \[(2,3)\]

and radius 5, the furthest distance is \[\sqrt{2^2+3^2}+5=\sqrt{13}+5\]

.The point is further from the origin than the centre of the circle, along the same line, by a factor

\[\frac{\sqrt{13}+5}{\sqrt{13}}\]

.Hence the coordinate of the point is

\[\frac{\sqrt{13}+5}{\sqrt{13}}(2,3)=\frac{13+5 \sqrt{13}}{13} (2,3)=(\frac{26+10 \sqrt{13}}{13}, \frac{39+15 \sqrt{13}}{13})\]

.