To determine the equation of a circle given its center and radius, we'll use the standard form of a circle's equation. The general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- The center of the circle: [tex]\((-3, -5)\)[/tex]
- The radius of the circle: [tex]\(4\)[/tex]
We can identify the values:
- [tex]\( h = -3 \)[/tex]
- [tex]\( k = -5 \)[/tex]
- [tex]\( r = 4 \)[/tex]
First, we substitute [tex]\( h, k, \)[/tex] and [tex]\( r \)[/tex] into the standard equation.
[tex]\[ (x - (-3))^2 + (y - (-5))^2 = 4^2 \][/tex]
Simplifying the terms inside the parentheses:
[tex]\[ (x + 3)^2 + (y + 5)^2 = 4^2 \][/tex]
Next, we calculate [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Thus, the equation of the circle becomes:
[tex]\[ (x + 3)^2 + (y + 5)^2 = 16 \][/tex]
Now we compare this result with the provided choices:
A. [tex]\((x-3)^2+(y-5)^2=16\)[/tex]
B. [tex]\((x+3)^2+(y+5)^2=16\)[/tex]
C. [tex]\((x-3)^2+(y-5)^2=4\)[/tex]
D. [tex]\((x+3)^2+(y+5)^2=4\)[/tex]
The correct equation is:
B. [tex]\((x+3)^2+(y+5)^2=16\)[/tex]
Therefore, the correct choice is B.
