Key Concepts and Formulas

• Equation of a Circle: A circle with center $(h, k)$ and radius $r$ has the equation $(x-h)^2 + (y-k)^2 = r^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Condition for a Circle to Touch the x-axis: If a circle with center $(h, k)$ touches the x-axis, its radius is $|k|$.
• Condition for External Tangency: Two circles with centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$ touch externally if the distance between their centers is equal to the sum of their radii: $C_1C_2 = r_1 + r_2$.

Step-by-Step Solution

Step 1: Identify the center and radius of the given circle.

The given circle has the equation $x^2 + y^2 - 8x - 8y - 4 = 0$. We can rewrite this in standard form by completing the square: $(x^2 - 8x) + (y^2 - 8y) = 4$ $(x^2 - 8x + 16) + (y^2 - 8y + 16) = 4 + 16 + 16$ $(x - 4)^2 + (y - 4)^2 = 36 = 6^2$ Thus, the given circle has center $C_1 = (4, 4)$ and radius $r_1 = 6$.

Step 2: Define the center and radius of the variable circle.

Let the center of the circle that touches the given circle externally and also touches the x-axis be $C_2 = (h, k)$ and its radius be $r_2$. Since this circle touches the x-axis, its radius must be equal to the absolute value of its y-coordinate: $r_2 = |k|$. Since the circle touches the given circle externally, we can assume that $k>0$, and thus $r_2 = k$.

Step 3: Apply the condition for external tangency.

The distance between the centers of the two circles must be equal to the sum of their radii. Therefore, $C_1C_2 = r_1 + r_2$ $\sqrt{(h - 4)^2 + (k - 4)^2} = 6 + k$ Squaring both sides, we get $(h - 4)^2 + (k - 4)^2 = (6 + k)^2$ $(h - 4)^2 + k^2 - 8k + 16 = 36 + 12k + k^2$ $(h - 4)^2 = 12k + 8k + 36 - 16$ $(h - 4)^2 = 20k + 20$ $(h - 4)^2 = 20(k + 1)$

Step 4: Find the locus of the center of the variable circle.

To find the locus, replace $h$ with $x$ and $k$ with $y$: $(x - 4)^2 = 20(y + 1)$ This is the equation of a parabola.

Common Mistakes & Tips

• Remember to complete the square correctly when finding the center and radius of the given circle.
• When applying the condition for external tangency, ensure you are adding the radii, not subtracting.
• Be careful with signs when squaring the distance formula.

Summary

We found the center and radius of the given circle by completing the square. Then, we defined the center and radius of the variable circle that touches the given circle externally and also touches the x-axis. By using the condition for external tangency and the fact that the radius of the variable circle equals the absolute value of its y-coordinate, we derived the equation of the locus of the center of the variable circle, which turned out to be a parabola.

Final Answer

The final answer is \boxed{a parabola}, which corresponds to option (D).