To determine which equation represents the circle with the given center and diameter, we will follow a step-by-step approach:
Step 1: Identify the center and radius
- The center of the circle is given as [tex]\((-4, 9)\)[/tex].
- The diameter of the circle is given as 10 units.
To find the radius, we use the formula for the radius:
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} \][/tex]
[tex]\[ \text{Radius} = \frac{10}{2} = 5 \][/tex]
Step 2: Write the standard form of the circle's equation
The standard form of the circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
For our circle:
- The center is [tex]\((-4, 9)\)[/tex], so [tex]\(h = -4\)[/tex] and [tex]\(k = 9\)[/tex].
- The radius is [tex]\(5\)[/tex], so [tex]\(r^2 = 5^2 = 25\)[/tex].
Substituting these values into the standard form, we get:
[tex]\[ (x + 4)^2 + (y - 9)^2 = 25 \][/tex]
Step 3: Compare with the given equations
Let's go through the given equations and see which one matches our derived equation:
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]
From the equations above, we identify that:
- Equation 2: [tex]\((x + 4)^2 + (y - 9)^2 = 25\)[/tex] matches our derived equation.
Therefore, the correct equation that represents the circle with center [tex]\((-4, 9)\)[/tex] and a diameter of 10 units is:
[tex]\[ \boxed{(x + 4)^2 + (y - 9)^2 = 25} \][/tex]
