To determine which equation represents a circle with a center at [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units, we need to 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.
### Step-by-Step Solution:
1. Identify the center [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]:
- The center [tex]\((h, k)\)[/tex] is given as [tex]\((-3, 5)\)[/tex].
- The radius [tex]\(r\)[/tex] is given as 4 units.
2. Substitute the values into the standard equation:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = 5\)[/tex]
- [tex]\(r = 4\)[/tex]
3. Write the equation with the given center and radius:
- Substitute [tex]\(h = -3\)[/tex] and [tex]\(k = 5\)[/tex] into the equation:
[tex]\[ (x - (-3))^2 + (y - 5)^2 = 4^2 \][/tex]
- Simplify the terms:
[tex]\[ (x + 3)^2 + (y - 5)^2 = 16 \][/tex]
4. Match the derived equation with the given options:
Let's look at each of the options provided:
A. [tex]\((x-3)^2 + (y+5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared would be 4, which does not match our parameters.
B. [tex]\((x-3)^2 + (y+5)^2 = 16\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared is 16. Though the radius squared is correct, the center does not match our parameters.
C. [tex]\((x+3)^2 + (y-5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((-3, 5)\)[/tex] but the radius squared is 4, which does not match our parameters.
D. [tex]\((x+3)^2 + (y-5)^2 = 16\)[/tex]
- Here, the center is [tex]\((-3, 5)\)[/tex] and the radius squared is 16. This perfectly matches our derived equation.
The correct answer is:
[tex]\[ \boxed{(x+3)^2 + (y-5)^2 = 16} \][/tex]
Thus, the equation representing a circle with center [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units is:
D. [tex]\((x+3)^2+(y-5)^2=16\)[/tex]
