Question

find the center and the radius of the following circle. x² - 12x + y² - 16y = 0 the center is (type an ordered pair.) the radius is (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 use the formula $(a - b)^2=a^2-2ab + b^2$. Here $a = x$ and $2b = 12$ (so $b = 6$), and $x^2-12x=(x - 6)^2-36$.

Step2: Complete the square for y - terms

For the $y^2-16y$ part, using the same formula with $a = y$ and $2b = 16$ (so $b = 8$), we get $y^2-16y=(y - 8)^2-64$.

Step3: Rewrite the circle equation

The original equation $x^2-12x + y^2-16y=0$ becomes $(x - 6)^2-36+(y - 8)^2-64 = 0$.

Step4: Simplify the equation

Rearranging, we have $(x - 6)^2+(y - 8)^2=36 + 64$.

Step5: 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, $(x - 6)^2+(y - 8)^2 = 100$, the center is $(6,8)$ and the radius $r=\sqrt{100}=10$.

Answer:

The center is $(6,8)$.
The radius is $10$.