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Step-by-Step Solution
Step 1: Identify the center and radius of the given circle
The circle given is
$x^2 + y^2 + 2x - 4y - 4 = 0.$
We can rewrite it as
$(x^2 + 2x) + (y^2 - 4y) = 4.$
Completing the square for each variable:
$x^2 + 2x + 1 - 1 + y^2 - 4y + 4 - 4 = 4,$
$(x + 1)^2 - 1 + (y - 2)^2 - 4 = 4.$
So,
$(x + 1)^2 + (y - 2)^2 = 9.$
This shows the center of the given circle is $(-1,\,2)$ and its radius is $3.$
Step 2: Understand the external tangential condition
Circle $C$, also of radius $3,$ touches the given circle externally at the point $(2,\,2).$
When two circles of radii $r_1$ and $r_2$ touch externally, the distance between their centers is $r_1 + r_2.$
Here, $r_1 = 3$ and $r_2 = 3,$ so the distance between their centers is $3 + 3 = 6.$
Step 3: Find the center of the new circle $C$
Since they touch externally at $(2,\,2)$, the line connecting the two centers (from $(-1, 2)$ to the unknown center $(h, k)$) must pass through the tangential point $(2,\,2).$
The vector from $(-1,\,2)$ to $(2,\,2)$ is $(3,\,0).$ Observe that the point $(2,\,2)$ is $3$ units horizontally from $(-1,\,2).$
Because the total distance between centers is $6,$ we extend from $(-1,\,2)$ by $6$ units in the same direction.
Moving $6$ units to the right (horizontally) from $(-1,\,2)$ yields the new center $(5,\,2).$
Step 4: Write the equation of circle $C$
The circle $C$ has center $(5,\,2)$ and radius $3.$ Its equation is:
$(x - 5)^2 + (y - 2)^2 = 3^2.$
Expanding:
$x^2 - 10x + 25 + y^2 - 4y + 4 = 9,$
$x^2 + y^2 - 10x - 4y + 20 = 0.$
Step 5: Determine the intercept on the x-axis
To find the x-intercepts, we set $y = 0$ in the equation of circle $C$:
$x^2 - 10x + 20 = 0.$
The roots are found by the quadratic formula:
$x = \frac{10 \pm \sqrt{(10)^2 - 4 \cdot 1 \cdot 20}}{2}
= \frac{10 \pm \sqrt{100 - 80}}{2}
= \frac{10 \pm \sqrt{20}}{2}
= 5 \pm \sqrt{5}.$
Hence, the two x-intercepts are $5 - \sqrt{5}$ and $5 + \sqrt{5}.$
Step 6: Calculate the length of the intercept
The length of the intercept on the x-axis is the difference between these two intercepts:
$\bigl(5 + \sqrt{5}\bigr) - \bigl(5 - \sqrt{5}\bigr) = 2\sqrt{5}.$
Answer: $2\sqrt{5}$
