Answer :
To determine which equations represent circles with a diameter of 12 units and a center that lies on the y-axis, let's break down the problem step-by-step.
### Step 1: Understand the Characteristics of the Circle
- Diameter: 12 units.
- Radius: The radius [tex]\( r \)[/tex] is half of the diameter, so [tex]\( r = \frac{12}{2} = 6 \)[/tex] units.
- Standard Form of a Circle's Equation: [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.
- The center must lie on the [tex]\( y \)[/tex]-axis, meaning the x-coordinate of the center must be 0. Therefore, the equation will have the form [tex]\(x^2 + (y - k)^2 = r^2\)[/tex], with [tex]\(h = 0\)[/tex].
### Step 2: Determine the Equation Parameters
- Radius Squared: The radius is 6, so the radius squared [tex]\( r^2 = 6^2 = 36 \)[/tex].
### Step 3: Check Each Given Equation to See if it Matches the Requirements
1. Equation: [tex]\(x^2 + (y - 3)^2 = 36\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, 3)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This equation matches the requirements.
2. Equation: [tex]\(x^2 + (y - 5)^2 = 6\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, 5)\)[/tex]
- Radius squared: [tex]\(6\)[/tex]
- This equation does not match because the radius squared is not 36.
3. Equation: [tex]\((x - 4)^2 + y^2 = 36\)[/tex]
- Form: [tex]\((x - h)^2 + y^2 = r^2\)[/tex]
- Center: [tex]\((4, 0)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This does not match because the center is not on the y-axis.
4. Equation: [tex]\((x + 6)^2 + y^2 = 144\)[/tex]
- Form: [tex]\((x - h)^2 + y^2 = r^2\)[/tex]
- Center: [tex]\((-6, 0)\)[/tex]
- Radius squared: [tex]\(144\)[/tex]
- This does not match because the radius squared is not 36 and the center is not on the y-axis.
5. Equation: [tex]\(x^2 + (y + 8)^2 = 36\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, -8)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This equation matches the requirements.
### Conclusion
The two equations that represent circles with a diameter of 12 units and centered on the y-axis are:
1. [tex]\(x^2 + (y - 3)^2 = 36\)[/tex]
2. [tex]\(x^2 + (y + 8)^2 = 36\)[/tex]
### Step 1: Understand the Characteristics of the Circle
- Diameter: 12 units.
- Radius: The radius [tex]\( r \)[/tex] is half of the diameter, so [tex]\( r = \frac{12}{2} = 6 \)[/tex] units.
- Standard Form of a Circle's Equation: [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.
- The center must lie on the [tex]\( y \)[/tex]-axis, meaning the x-coordinate of the center must be 0. Therefore, the equation will have the form [tex]\(x^2 + (y - k)^2 = r^2\)[/tex], with [tex]\(h = 0\)[/tex].
### Step 2: Determine the Equation Parameters
- Radius Squared: The radius is 6, so the radius squared [tex]\( r^2 = 6^2 = 36 \)[/tex].
### Step 3: Check Each Given Equation to See if it Matches the Requirements
1. Equation: [tex]\(x^2 + (y - 3)^2 = 36\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, 3)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This equation matches the requirements.
2. Equation: [tex]\(x^2 + (y - 5)^2 = 6\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, 5)\)[/tex]
- Radius squared: [tex]\(6\)[/tex]
- This equation does not match because the radius squared is not 36.
3. Equation: [tex]\((x - 4)^2 + y^2 = 36\)[/tex]
- Form: [tex]\((x - h)^2 + y^2 = r^2\)[/tex]
- Center: [tex]\((4, 0)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This does not match because the center is not on the y-axis.
4. Equation: [tex]\((x + 6)^2 + y^2 = 144\)[/tex]
- Form: [tex]\((x - h)^2 + y^2 = r^2\)[/tex]
- Center: [tex]\((-6, 0)\)[/tex]
- Radius squared: [tex]\(144\)[/tex]
- This does not match because the radius squared is not 36 and the center is not on the y-axis.
5. Equation: [tex]\(x^2 + (y + 8)^2 = 36\)[/tex]
- Form: [tex]\(x^2 + (y - k)^2 = r^2\)[/tex]
- Center: [tex]\((0, -8)\)[/tex]
- Radius squared: [tex]\(36\)[/tex]
- This equation matches the requirements.
### Conclusion
The two equations that represent circles with a diameter of 12 units and centered on the y-axis are:
1. [tex]\(x^2 + (y - 3)^2 = 36\)[/tex]
2. [tex]\(x^2 + (y + 8)^2 = 36\)[/tex]
