Question

on a coordinate plane, a circle has the equation x² + y² - 6y - 16 = 0. which ordered pair describes the center of the circle?
a) (0, 3)
b) (0, 5)
c) (3, 5)
d) (6, 16)

Explanation:

Step1: Rewrite the circle equation in 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. Given $x^{2}+y^{2}-6y - 16=0$. Complete the square for the $y$ - terms. We have $x^{2}+y^{2}-6y+9-9 - 16=0$. Then $x^{2}+(y - 3)^{2}-25=0$, which can be rewritten as $x^{2}+(y - 3)^{2}=25$.

Step2: Identify the center of the circle

Comparing $x^{2}+(y - 3)^{2}=25$ with the standard - form $(x - a)^2+(y - b)^2=r^2$, we can see that $a = 0$ and $b = 3$. So the center of the circle is $(0,3)$.

Answer:

A. (0, 3)