# The center of a circle is at (0,0) and its radius is 5. Does the point (5,-2) lie on the circle?

Jan 18, 2016

No

#### Explanation:

A circle with center $c$ and radius $r$ is the locus (collection) of points which are distance $r$ from $c$. Thus, given $r$ and $c$, we can tell if a point is on the circle by seeing if it is distance $r$ from $c$.

The distance between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ can be calculated as
$\text{distance} = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$
(This formula can be derived using the Pythagorean theorem)

So, the distance between $\left(0 , 0\right)$ and $\left(5 , - 2\right)$ is
$\sqrt{{\left(5 - 0\right)}^{2} + {\left(- 2 - 0\right)}^{2}} = \sqrt{25 + 4} = \sqrt{29}$

As $\sqrt{29} \ne 5$ this means that $\left(5 , - 2\right)$ does not lie on the given circle.