Question

find the center and the radius of the following circle.
x² - 12x + y² - 10y = 0
the center is (6,5). (type an ordered - pair.)
the radius is 10. (simplify your answer.)
use the graphing tool to graph the circle.
click to enlarge graph

Explanation:

Step1: Complete the square for x - terms

For the $x^{2}-12x$ part, we know that $(x - a)^2=x^{2}-2ax + a^{2}$. Here $-2a=-12$, so $a = 6$ and $x^{2}-12x=(x - 6)^{2}-36$.

Step2: Complete the square for y - terms

For the $y^{2}-10y$ part, since $-2b=-10$, then $b = 5$ and $y^{2}-10y=(y - 5)^{2}-25$.

Step3: Rewrite the circle equation

The original equation $x^{2}-12x + y^{2}-10y=0$ becomes $(x - 6)^{2}-36+(y - 5)^{2}-25 = 0$, which simplifies to $(x - 6)^{2}+(y - 5)^{2}=36 + 25$.

Step4: Identify the center and radius

The standard - form of a circle equation is $(x - h)^{2}+(y - k)^{2}=r^{2}$, where $(h,k)$ is the center and $r$ is the radius. So the center is $(6,5)$ and $r^{2}=61$, then $r=\sqrt{61}$.

Answer:

The center is $(6,5)$; The radius is $\sqrt{61}$