Question

in the xy - plane, the graph of the given equation is circle. what are the coordinates (x,y) of the center of the circle? x² + 44x + y² = 0
a. (-22,0)
b. (22,0)
c. (0,-22)
d. (0,22)

Explanation:

Step1: Recall circle - standard form

The general equation of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle. For the given equation $x^{2}+44x + y^{2}=0$, we complete the square for the $x$ - terms.

Step2: Complete the square for $x$

We know that $x^{2}+44x=(x + 22)^2-484$. So the equation $x^{2}+44x + y^{2}=0$ can be rewritten as $(x + 22)^2-484+y^{2}=0$, which simplifies to $(x + 22)^2+y^{2}=484$.

Step3: Identify the center

Comparing $(x + 22)^2+y^{2}=484$ with the standard - form $(x - a)^2+(y - b)^2=r^2$, we have $a=-22$ and $b = 0$. So the center of the circle is $(-22,0)$.

Answer:

a. (-22,0)