# How do you find the center and radius of the circle (x-2)^2 + (y-5)^2=100?

May 26, 2016

Center is at $\left(2 , 5\right)$, radius is $10$

#### Explanation:

Graph that corresponds to an equation ${\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {R}^{2}$ is a locus of points $\left(x , y\right)$ that transform this equation into identity.

Consider any point $\left(x , y\right)$ that does transform this equation into identity.
What is the distance from this point to $\left(a , b\right)$?

As we know, the distance between points $\left(x , y\right)$ and $\left(a , b\right)$ is $d = \sqrt{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2}}$. The expression on the right equals to $R$ since coordinates $\left(x , y\right)$ transform the equation ${\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {R}^{2}$ into an identity.

Therefore, the distance from $\left(x , y\right)$ to $\left(a , b\right)$ (we denoted it as $d$) is always equal to $R$.

As we see, the distance $d$ from any point satisfying the above equation (that is from any point of graph of this equation) to point $\left(a , b\right)$ is constant $R$. But that is a definition of a circle with a center $\left(a , b\right)$ and radius $R$.