Question

A circle with center $\left(0,0\right)$ passes through the point $\left(5,12\right)$.

How do you find the radius of the circle?

Answer 1

Finding the radius of the given circle:

Suppose the radius of the given circle is $r$.

We know that the equation of a circle of radius $r$ with the center at the point $\left(a,b\right)$ is given by: ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$.

Now, given that the center is at $\left(0,0\right)$.

So, putting $a=0,b=0$ in ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$, we get: $x^2+y^2=r^2$.

Also, given that the circle passes through the point $\left(5,12\right)$.

So, putting $x=5,y=12$ in $x^2+y^2=r^2$, we get:

$$\begin{gathered} 5^2+{12}^2=r^2 \\ \Rightarrow r^2={13}^2 \\ \therefore r=13 \end{gathered}$$

Therefore, the radius of the given circle is $r=13$ unit.