A circle `C_(1)` of radius 2 units rolls o the outerside of the circle `C_(2) : x^(2) + y^(2) + 4x = 0` touching it externally.
Square of the length of the intercept made by `x^(2) + y^(2) + 4x - 12 = 0` on any tangents to `C_(2)` is

Options

A. (a)12
B. (b)24
C. (c)16
D. (d)48

Answer

To solve the problem, we need to follow these steps:

### Step 1: Identify the center and radius of circle C2
The equation of circle C2 is given as:
\[ x^2 + y^2 + 4x = 0 \]

We can rewrite this in standard form:
\[ (x + 2)^2 + y^2 = 2^2 \]

From this, we can see that the center of circle C2 is:
\[ (-2, 0) \]
and the radius \( r_2 \) is:
\[ 2 \text{ units} \]

### Step 2: Identify the equation of the second circle
The equation of the second circle (C1) that rolls around C2 is:
\[ x^2 + y^2 + 4x - 12 = 0 \]

We can rewrite this in standard form:
\[ (x + 2)^2 + y^2 = 4^2 \]

From this, we can see that the center of circle C1 is also:
\[ (-2, 0) \]
and the radius \( r_1 \) is:
\[ 4 \text{ units} \]

### Step 3: Find the length of the intercept made by the tangents to circle C2
The length of the intercept made by tangents to circle C2 can be found using the formula for the length of the tangent from a point to a circle:
\[ L = \sqrt{d^2 - r^2} \]
where \( d \) is the distance from the center of the circle to the point, and \( r \) is the radius of the circle.

### Step 4: Calculate the distance from the center of C2 to the center of C1
The distance \( d \) from the center of C2 to the center of C1 is:
\[ d = r_2 + r_1 = 2 + 2 = 4 \text{ units} \]

### Step 5: Calculate the square of the length of the intercept
Using the formula for the length of the tangent:
\[ L^2 = d^2 - r_2^2 \]
Substituting the values:
\[ L^2 = 4^2 - 2^2 = 16 - 4 = 12 \]

### Step 6: Find the square of the length of the intercept
The square of the length of the intercept made by the tangents to circle C2 is:
\[ L^2 = 12 \]

### Final Answer
Thus, the square of the length of the intercept made by the tangents to circle C2 is:
\[ \boxed{12} \]

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