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

Suppose we have a circle and we want to find the point on the circle furthest from the origin. This point must lie on the straight line from the origin, through the centre of the circle, to the point on the circle on the opposite side of the centre of the circle 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
$\frac{\sqrt{13}+5}{\sqrt{13}}(2,3)=\frac{13+5 \sqrt{13}}{13} (2,3)=(\frac{26+10 \sqrt{13}}{13}, \frac{39+15 \sqrt{13}}{13})$
.