To determine the equation that represents a circle with a center at [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units, we use the standard form of the equation of a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\( (h, k) \)[/tex] is the center of the circle and [tex]\( r \)[/tex] is the radius.
Given in the problem:
- The center [tex]\( (h, k) \)[/tex] = [tex]\((-3, 5)\)[/tex]
- The radius [tex]\( r \)[/tex] = 4 units
We substitute [tex]\( h = -3 \)[/tex], [tex]\( k = 5 \)[/tex], and [tex]\( r = 4 \)[/tex] into the standard form equation:
[tex]\[
(x - (-3))^2 + (y - 5)^2 = 4^2
\][/tex]
Simplify the equation:
[tex]\[
(x + 3)^2 + (y - 5)^2 = 16
\][/tex]
Hence, the equation that represents the circle is:
[tex]\[
(x + 3)^2 + (y - 5)^2 = 16
\][/tex]
Now we match this with the provided choices:
A. [tex]\((x-3)^2+(y+5)^2=4\)[/tex]
B. [tex]\((x-3)^2+(y+5)^2=16\)[/tex]
C. [tex]\((x+3)^2+(y-5)^2=4\)[/tex]
D. [tex]\((x+3)^2+(y-5)^2=16\)[/tex]
The correct answer is:
D. [tex]\((x+3)^2+(y-5)^2=16\)[/tex]
