If a circle C, whose radius is 3, touches externally the circle, `x^(2) +y^(2)+2x - 4y - 4=0` at the point (2,2) ,then the length of intercept cut by this circle C, the x-axis is equal to `:`

Options

A. `2 sqrt(3)`
B. `3sqrt(2)`
C. ` sqrt( 5)`
D. `2 sqrt( 5)`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Identify the given circle's equation
The equation of the given circle is:
\[ x^2 + y^2 + 2x - 4y - 4 = 0 \]

### Step 2: Rewrite the circle's equation in standard form
To rewrite the equation in standard form, we complete the square for both \(x\) and \(y\).

1. Rearranging the equation:
\[ x^2 + 2x + y^2 - 4y = 4 \]

2. Completing the square:
- For \(x\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
- For \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

3. Substituting back:
\[ (x + 1)^2 - 1 + (y - 2)^2 - 4 = 4 \]
\[ (x + 1)^2 + (y - 2)^2 = 9 \]

This shows that the center of the given circle is \((-1, 2)\) and its radius is \(3\).

### Step 3: Determine the center of circle C
Since circle C touches the given circle externally at the point \((2, 2)\), the distance between the centers of the two circles is equal to the sum of their radii.

Let the center of circle C be \((P, Q)\). The distance between the centers is:
\[ \sqrt{(P + 1)^2 + (Q - 2)^2} = 3 + 3 = 6 \]

### Step 4: Set up equations based on the midpoint
Since the point \((2, 2)\) is the midpoint of the line segment joining the centers of the two circles, we can set up the following equations:
1. \(\frac{P - 1}{2} = 2\)
2. \(\frac{Q + 2}{2} = 2\)

From these, we can solve for \(P\) and \(Q\):
1. \(P - 1 = 4 \Rightarrow P = 5\)
2. \(Q + 2 = 4 \Rightarrow Q = 2\)

Thus, the center of circle C is \((5, 2)\).

### Step 5: Write the equation of circle C
The equation of circle C with center \((5, 2)\) and radius \(3\) is:
\[ (x - 5)^2 + (y - 2)^2 = 3^2 \]
Expanding this gives:
\[ (x - 5)^2 + (y - 2)^2 = 9 \]
\[ x^2 - 10x + 25 + y^2 - 4y + 4 = 9 \]
\[ x^2 + y^2 - 10x - 4y + 20 = 0 \]

### Step 6: Find the x-intercept of circle C
To find the x-intercept, set \(y = 0\) in the equation of circle C:
\[ x^2 + 0^2 - 10x - 4(0) + 20 = 0 \]
This simplifies to:
\[ x^2 - 10x + 20 = 0 \]

### Step 7: Calculate the x-intercepts using the quadratic formula
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = -10\), and \(c = 20\):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} \]
\[ x = \frac{10 \pm \sqrt{100 - 80}}{2} \]
\[ x = \frac{10 \pm \sqrt{20}}{2} \]
\[ x = \frac{10 \pm 2\sqrt{5}}{2} \]
\[ x = 5 \pm \sqrt{5} \]

### Step 8: Find the length of the intercept on the x-axis
The length of the intercept is the distance between the two x-intercepts:
\[ \text{Length} = (5 + \sqrt{5}) - (5 - \sqrt{5}) = 2\sqrt{5} \]

### Final Answer
Thus, the length of the intercept cut by circle C on the x-axis is:
\[ \boxed{2\sqrt{5}} \]