To solve the problem, we need to identify which given equations represent circles with a diameter of 12 units and a center that lies on the [tex]$y$[/tex]-axis. We will analyze each equation step-by-step.
A general circle equation is given by:
[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 a circle with a diameter of 12 units, the radius \( r \) is half of the diameter:
[tex]\[r = \frac{12}{2} = 6\][/tex]
So, the radius squared (\( r^2 \)) is:
[tex]\[r^2 = 6^2 = 36\][/tex]
### Analysis of Each Equation
1. Equation: \( x^2 + (y - 3)^2 = 36 \)
- The center is at \((0, 3)\). Since the center lies on the [tex]$y$[/tex]-axis and \( r^2 = 36 \) matches our requirement, this equation is valid.
2. Equation: \( x^2 + (y - 5)^2 = 6 \)
- The center is at \((0, 5)\). While the center lies on the [tex]$y$[/tex]-axis, \( r^2 = 6 \) does not satisfy the radius condition because the radius squared should be 36. This equation is not valid.
3. Equation: \( (x - 1)^2 + 30 = 30 \)
- Rearranging to the general circle form \((x - h)^2 + (y - k)^2 = r^2\), we get \((x - 1)^2 + 0 = 0\). This indicates that it's not a valid circle equation. Thus, it's not valid.
4. Equation: \( (x + 6)^2 + y^2 = 144 \)
- The center is at \((-6, 0)\). The center does not lie on the [tex]$y$[/tex]-axis, so this equation is not valid.
5. Equation: \( x^2 + (y + 8)^2 = 36 \)
- The center is at \((0, -8)\). Since the center lies on the [tex]$y$[/tex]-axis and \( r^2 = 36 \) matches our requirement, this equation is valid.
### Conclusion
The equations that represent circles with a diameter of 12 units and a center that lies on the [tex]$y$[/tex]-axis are:
[tex]\[x^2 + (y - 3)^2 = 36 \][/tex]
and
[tex]\[x^2 + (y + 8)^2 = 36.\][/tex]
Therefore, the correct selections are:
[tex]\[1. \quad x^2 + (y - 3)^2 = 36 \][/tex]
[tex]\[5. \quad x^2 + (y + 8)^2 = 36.\][/tex]
Thus, the two options you should select are:
1. \( x^2 + (y - 3)^2 = 36 \)
2. [tex]\( x^2 + (y + 8)^2 = 36 \)[/tex]
