To determine the possible equations of the circle given its diameter and the fact that its center lies on the [tex]\( x \)[/tex]-axis, we need to follow these steps:
1. Determine the radius:
- The diameter of the circle is 12 units.
- The radius of the circle is half of the diameter, so:
[tex]\[
\text{Radius} = \frac{12}{2} = 6 \text{ units}
\][/tex]
2. Determine the possible coordinates of the center of the circle:
- Since the center lies on the [tex]\( x \)[/tex]-axis, its [tex]\( y \)[/tex]-coordinate is 0.
- The [tex]\( x \)[/tex]-coordinate can be any value on the [tex]\( x \)[/tex]-axis. Given the radius is 6, the center [tex]\( (h, k) \)[/tex] can be at:
[tex]\[
(6, 0) \quad \text{or} \quad (-6, 0)
\][/tex]
3. Formulate the equation of the circle using the standard form:
- The general 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]
4. Substitute the centers and radius into the equation:
- For the center at [tex]\( (6, 0) \)[/tex]:
[tex]\[
(x - 6)^2 + y^2 = 6^2 \implies (x - 6)^2 + y^2 = 36
\][/tex]
- For the center at [tex]\( (-6, 0) \)[/tex]:
[tex]\[
(x + 6)^2 + y^2 = 6^2 \implies (x + 6)^2 + y^2 = 36
\][/tex]
5. Compare with the given options:
- Option 1: [tex]\( (x-12)^2 + y^2 = 12 \)[/tex]
- This does not match either of our equations.
- Option 2: [tex]\( (x-6)^2 + y^2 = 36 \)[/tex]
- This matches the equation when the center is at [tex]\( (6, 0) \)[/tex].
- Option 3: [tex]\( x^2 + y^2 = 12 \)[/tex]
- This does not match either of our equations.
- Option 4: [tex]\( x^2 + y^2 = 144 \)[/tex]
- This does not match either of our equations.
- Option 5: [tex]\( (x+6)^2 + y^2 = 36 \)[/tex]
- This matches the equation when the center is at [tex]\( (-6, 0) \)[/tex].
- Option 6: [tex]\( (x+12)^2 + y^2 = 144 \)[/tex]
- This does not match either of our equations.
Thus, the correct equations of the circle are:
[tex]\[
(x-6)^2 + y^2 = 36
\][/tex]
and
[tex]\[
(x+6)^2 + y^2 = 36
\][/tex]
So, the correct options are:
- Option 2: [tex]\( (x-6)^2 + y^2 = 36 \)[/tex]
- Option 5: [tex]\( (x+6)^2 + y^2 = 36 \)[/tex]
Therefore, the correct answers are:
[tex]\[
\boxed{[2, 5]}
\][/tex]
