Key Concepts and Formulas

• Equation of a circle with center $(h, k)$ and radius $r$: $(x-h)^2 + (y-k)^2 = r^2$
• Distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Condition for two circles to touch externally: The distance between their centers is equal to the sum of their radii.

Step-by-Step Solution

Step 1: Define the fixed circle.

We are given a circle with center $(0, 3)$ and radius $2$. Let's call this circle $C_1$.

• Center of $C_1$: $(0, 3)$
• Radius of $C_1$: $r_1 = 2$ The equation of $C_1$ is $x^2 + (y-3)^2 = 4$.

Step 2: Define the variable circle.

Let the center of the variable circle be $(h, k)$. Let's call this circle $C_2$.

• Center of $C_2$: $(h, k)$

This variable circle touches the x-axis. This means its radius is equal to the absolute value of its y-coordinate, which is $|k|$. Since we are looking for a locus, we can assume $k > 0$ (the circle is above the x-axis), so the radius is simply $k$.

• Radius of $C_2$: $r_2 = k$ The equation of $C_2$ is $(x-h)^2 + (y-k)^2 = k^2$.

Step 3: Apply the condition for external touching.

The variable circle $C_2$ touches the fixed circle $C_1$ externally. This means the distance between their centers is equal to the sum of their radii.

• Distance between centers $(0, 3)$ and $(h, k)$: $C_1C_2 = \sqrt{(h-0)^2 + (k-3)^2} = \sqrt{h^2 + (k-3)^2}$
• Sum of radii: $r_1 + r_2 = 2 + k$
• Equating the distance to the sum of radii: $\sqrt{h^2 + (k-3)^2} = 2 + k$

Step 4: Simplify the equation to find the relationship between $h$ and $k$.

Square both sides of the equation to eliminate the square root: $h^2 + (k-3)^2 = (2+k)^2$ Expand the terms: $h^2 + k^2 - 6k + 9 = 4 + 4k + k^2$ Simplify the equation: $h^2 - 6k + 9 = 4 + 4k$ $h^2 = 10k - 5$ $h^2 = 10(k - \frac{1}{2})$

Step 5: Identify the locus.

Replacing $h$ with $x$ and $k$ with $y$, we get the equation of the locus: $x^2 = 10(y - \frac{1}{2})$ This is the equation of a parabola.

Common Mistakes & Tips

• Remember to square both sides of the equation carefully to eliminate the square root.
• The radius of a circle touching the x-axis is the absolute value of the y-coordinate of its center.
• When dealing with loci problems, replace the coordinates of the variable point (here $(h, k)$) with $(x, y)$ at the end.

Summary

We found the locus of the center of a circle that touches the x-axis and another fixed circle. We defined the fixed circle and the variable circle, then used the condition for external touching to relate their centers and radii. After simplification, we obtained the equation $x^2 = 10(y - \frac{1}{2})$, which represents a parabola.

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