Key Concepts and Formulas

• Equation of a Circle: The equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$. The general form is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.
• External Tangency: If two circles with centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$ touch externally, the distance between their centers is $C_1C_2 = r_1 + r_2$, and the point of tangency lies on the line segment connecting the centers.
• X-intercept Length: The length of the intercept cut by a circle on the x-axis is given by $2\sqrt{r^2 - k^2}$, where $(h, k)$ is the center and $r$ is the radius. Equivalently, using the general form, it is $2\sqrt{g^2 - c}$.

Step-by-Step Solution

Step 1: Analyze the Given Circle

We are given the equation of the first circle as $x^2 + y^2 + 2x - 4y - 4 = 0$. We need to find its center and radius. Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = 2$, $2f = -4$, and $c = -4$. Therefore, $g = 1$, $f = -2$, and $c = -4$.

• The center of the first circle, $C_1$, is $(-g, -f) = (-1, 2)$.
• The radius of the first circle, $r_1$, is $\sqrt{g^2 + f^2 - c} = \sqrt{1^2 + (-2)^2 - (-4)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$.

So, the first circle has center $C_1(-1, 2)$ and radius $r_1 = 3$.

Step 2: Determine the Center of Circle C

Let the circle C be denoted as $C_2$. We are given that its radius $r_2 = 3$. Circle $C_2$ touches circle $C_1$ externally at the point $P(2, 2)$. Since the circles touch externally, the distance between their centers is the sum of their radii: $C_1C_2 = r_1 + r_2 = 3 + 3 = 6$.

Since $r_1 = r_2 = 3$, the point of contact $P(2, 2)$ is the midpoint of the line segment connecting the centers $C_1$ and $C_2$. Let the center of circle C be $C_2(h, k)$. Using the midpoint formula:

$2 = \frac{-1 + h}{2} \Rightarrow 4 = -1 + h \Rightarrow h = 5$ $2 = \frac{2 + k}{2} \Rightarrow 4 = 2 + k \Rightarrow k = 2$

So, the center of circle C, $C_2$, is $(5, 2)$.

Step 3: Write the Equation of Circle C

Now we have the center of circle C as $(h, k) = (5, 2)$ and its radius $r_2 = 3$. Using the standard equation of a circle $(x-h)^2 + (y-k)^2 = r^2$:

$(x-5)^2 + (y-2)^2 = 3^2$ $(x-5)^2 + (y-2)^2 = 9$

Expanding this equation to the general form:

$x^2 - 10x + 25 + y^2 - 4y + 4 = 9$ $x^2 + y^2 - 10x - 4y + 20 = 0$

This is the equation of circle C.

Step 4: Calculate the Length of the Intercept Cut by Circle C on the X-axis

We need to find the length of the x-intercept for the circle $x^2 + y^2 - 10x - 4y + 20 = 0$. Using the general form, we have $2g = -10 \Rightarrow g = -5$ and $c = 20$.

The formula for the length of the x-intercept is $2\sqrt{g^2 - c}$. Length $= 2\sqrt{(-5)^2 - 20}$ Length $= 2\sqrt{25 - 20}$ Length $= 2\sqrt{5}$

Alternatively, using the center $(h,k) = (5,2)$ and radius $r=3$: The formula for the length of x intercept is $2\sqrt{r^2-k^2} = 2\sqrt{3^2 - 2^2} = 2\sqrt{9-4} = 2\sqrt{5}$.

Common Mistakes & Tips

• Remember to correctly apply the midpoint formula when the point of tangency is the midpoint of the centers.
• Ensure you use the correct formula for the x-intercept. It's $2\sqrt{r^2 - k^2}$ or $2\sqrt{g^2-c}$, not $2\sqrt{r^2 - h^2}$ or $2\sqrt{f^2 - c}$ (which is for the y-intercept).
• Always check if $r^2 - k^2$ (or $g^2 - c$) is non-negative before calculating the intercept length, to ensure the circle intersects the x-axis.

Summary

We found the center and radius of the given circle, used the external tangency condition to find the center of the second circle, derived the equation of the second circle, and finally calculated the length of the intercept it cuts on the x-axis. The length of the intercept cut by circle C on the x-axis is $2\sqrt{5}$.

The final answer is $\boxed{2\sqrt 5}$, which corresponds to option (A).