To determine which equation represents a circle with a given center and diameter, we follow these steps:
1. Identify the center coordinates: The center of the circle is [tex]\((-4, 9)\)[/tex].
2. Determine the radius: The radius of a circle is half the diameter. The diameter given is 10 units.
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 \text{ units}
\][/tex]
3. Calculate the square of the radius: This is needed for the standard equation of a circle.
[tex]\[
\text{Radius squared} = 5^2 = 25
\][/tex]
4. Form the equation of the circle: The standard form of the 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]
Substituting [tex]\(h = -4\)[/tex], [tex]\(k = 9\)[/tex], and [tex]\(r^2 = 25\)[/tex], the equation becomes:
[tex]\[
(x - (-4))^2 + (y - 9)^2 = 25
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
(x + 4)^2 + (y - 9)^2 = 25
\][/tex]
Now, we compare this derived equation with the given choices:
1. [tex]\((x-9)^2+(y+4)^2=25\)[/tex]
2. [tex]\((x+4)^2+(y-9)^2=25\)[/tex]
3. [tex]\((x-9)^2+(y+4)^2=100\)[/tex]
4. [tex]\((x+4)^2+(y-9)^2=100\)[/tex]
The correct equation is:
[tex]\[
(x + 4)^2 + (y - 9)^2 = 25
\][/tex]
Thus, the correct choice is:
[tex]\[
(x + 4)^2 + (y - 9)^2 = 25
\][/tex]
