Question

give the center and radius of the circle represented by the equation. x^2 + y^2 + 10x + 4y + 20 = 0 the center of the circle is (type an ordered pair, using integers or fractions.) the radius of the circle is (type an integer or a simplified fraction.)

Explanation:

Step1: Complete the square for x - terms

The x - terms are $x^{2}+10x$. Completing the square: $x^{2}+10x=(x + 5)^{2}-25$.

Step2: Complete the square for y - terms

The y - terms are $y^{2}+4y$. Completing the square: $y^{2}+4y=(y + 2)^{2}-4$.

Step3: Rewrite the circle equation

Substitute the completed - square expressions into the original equation:
\[

$$\begin{align*} x^{2}+y^{2}+10x + 4y+20&=0\\ (x + 5)^{2}-25+(y + 2)^{2}-4+20&=0\\ (x + 5)^{2}+(y + 2)^{2}&=9 \end{align*}$$

\]

Step4: Identify the center and radius

The standard form of a circle equation is $(x - a)^{2}+(y - b)^{2}=r^{2}$, where $(a,b)$ is the center and $r$ is the radius.
For the equation $(x + 5)^{2}+(y + 2)^{2}=9=(x-(-5))^{2}+(y - (-2))^{2}$, the center is $(-5,-2)$ and the radius $r = 3$.

Answer:

The center of the circle is $(-5,-2)$
The radius of the circle is $3$