The centres of those circles which touch the circle, `x^2+y^2-8x-8y-4=0` , externally and also touch the x-axis, lie on : (1) a circle. (2) an ellipse which is not a circle. (3) a hyperbola. (4) a parabola.

Options

A. an ellipse which is not a circle
B. a hyperbola
C. a parabola
D. a circle

Explanation

To solve the problem, we need to find the locus of the centers of circles that touch the given circle externally and also touch the x-axis.

### Step-by-step Solution:

1. **Identify the given circle**:
The equation of the circle is \( x^2 + y^2 - 8x - 8y - 4 = 0 \). We can rewrite this in standard form by completing the square.

\[ (x^2 - 8x) + (y^2 - 8y) = 4 \]
Completing the square for \(x\) and \(y\):
\[ (x - 4)^2 - 16 + (y - 4)^2 - 16 = 4 \]
\[ (x - 4)^2 + (y - 4)^2 = 36 \]
This shows that the center of the circle is at \( (4, 4) \) and the radius \( r = 6 \).

2. **Let the center of the new circle be \( (h, k) \)**:
The equation of the new circle can be written as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

3. **Condition for external tangency**:
For the new circle to touch the given circle externally, the distance between the centers must equal the sum of the radii. Thus:
\[ \sqrt{(h - 4)^2 + (k - 4)^2} = r + 6 \]
where \( r \) is the radius of the new circle.

4. **Condition for touching the x-axis**:
If the new circle touches the x-axis, then the distance from the center to the x-axis must equal the radius \( r \). Therefore, we have:
\[ k = r \]

5. **Substituting \( k \)**:
We can substitute \( k \) in the equation from step 3:
\[ \sqrt{(h - 4)^2 + (r - 4)^2} = r + 6 \]
Squaring both sides:
\[ (h - 4)^2 + (r - 4)^2 = (r + 6)^2 \]

6. **Expanding and simplifying**:
Expanding both sides:
\[ (h - 4)^2 + (r^2 - 8r + 16) = (r^2 + 12r + 36) \]
Simplifying gives:
\[ (h - 4)^2 - 20r + 16 = 36 \]
\[ (h - 4)^2 = 20r + 20 \]

7. **Rearranging**:
Rearranging gives us:
\[ (h - 4)^2 = 20(k) + 20 \]
Replacing \( r \) with \( k \):
\[ (h - 4)^2 = 20k + 20 \]

8. **Final form**:
This can be rewritten as:
\[ (h - 4)^2 = 20(k + 1) \]
This is the equation of a parabola in the form \( (x - h_0)^2 = 4p(y - k_0) \).

### Conclusion:
The locus of the centers of the circles that touch the given circle externally and also touch the x-axis is a parabola.