To determine which equation represents a circle with a center at [tex]\((-4, 9)\)[/tex] and a diameter of 10 units, follow these steps:
1. Identify the center and radius of the circle:
- The center of the circle is given as [tex]\((-4, 9)\)[/tex].
- The diameter is given as 10 units. To find the radius, divide the diameter by 2:
[tex]\[
\text{Radius} = \frac{10}{2} = 5
\][/tex]
2. Write the standard equation of the circle:
- The general form of a circle's equation centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
- Here, [tex]\(h = -4\)[/tex] and [tex]\(k = 9\)[/tex], and the radius [tex]\(r = 5\)[/tex]. Substitute these values into the equation:
[tex]\[
(x + 4)^2 + (y - 9)^2 = 5^2
\][/tex]
- Simplify:
[tex]\[
(x + 4)^2 + (y - 9)^2 = 25
\][/tex]
3. Compare the derived equation with the given options to determine which match:
- Option 1: [tex]\((x - 9)^2 + (y + 4)^2 = 25\)[/tex]
- This equation does not match our derived equation because it has different signs and different center coordinates.
- Option 2: [tex]\((x + 4)^2 + (y - 9)^2 = 25\)[/tex]
- This matches our derived equation exactly.
- Option 3: [tex]\((x - 9)^2 + (y + 4)^2 = 100\)[/tex]
- This equation does not match because both the signs, coordinates and the right-hand side (should be 25) do not align with our derived equation.
- Option 4: [tex]\((x + 4)^2 + (y - 9)^2 = 100\)[/tex]
- This equation only partly matches, but the right-hand side (100) is incorrect.
Based on the comparison, the correct equation that represents the circle with a center at [tex]\((-4, 9)\)[/tex] and a diameter of 10 units is:
[tex]\[
\boxed{(x + 4)^2 + (y - 9)^2 = 25}
\][/tex]
