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.
Area of a quadrilateral found by a pair of tangents from a point of `x^(2) + y^(2) + 4x -12 = 0` to `C_(2)` with a pair of radii at the points of contact of the tangents is ( in sq. units )

Options

A. `2sqrt(3)`
B. `4sqrt(3)`
C. `sqrt(3)`
D. `3sqrt(3)`

Answer

To solve the problem, we need to find the area of a quadrilateral formed by tangents drawn from a point to a circle. Let's break down the solution step by step.

### Step 1: Identify the circles
The first circle \( C_1 \) has a radius of 2 units. The second circle \( C_2 \) is given by the equation:
\[ x^2 + y^2 + 4x = 0 \]
We can rewrite this equation by completing the square:
\[ (x + 2)^2 + y^2 = 4 \]
This shows that circle \( C_2 \) has a center at \( (-2, 0) \) and a radius of 2 units.

### Step 2: Determine the center and radius of the larger circle
Since circle \( C_1 \) rolls on the outside of circle \( C_2 \), the distance from the center of \( C_2 \) to the center of \( C_1 \) is the sum of their radii:
\[ \text{Distance} = 2 + 2 = 4 \text{ units} \]
Thus, the center of circle \( C_1 \) is located at \( (-2 + 4, 0) = (2, 0) \).

### Step 3: Write the equation of circle \( C_1 \)
The equation of circle \( C_1 \) can be written as:
\[ (x - 2)^2 + y^2 = 4 \]

### Step 4: Identify the point from which tangents are drawn
The point from which the tangents are drawn is given by the equation:
\[ x^2 + y^2 + 4x - 12 = 0 \]
Completing the square for this equation gives:
\[ (x + 2)^2 + y^2 = 16 \]
This shows that the point is at the center \( (-2, 0) \) with a radius of 4 units. The point from which the tangents are drawn is \( P(2, 0) \).

### Step 5: Calculate the area of the quadrilateral formed by the tangents
The area of the quadrilateral formed by the tangents from point \( P \) to circle \( C_2 \) can be calculated using the formula for the area of a quadrilateral formed by two tangents and the radii at the points of contact.

The length of each tangent from point \( P \) to circle \( C_2 \) can be calculated using the formula:
\[ \text{Length of tangent} = \sqrt{d^2 - r^2} \]
where \( d \) is the distance from point \( P \) to the center of circle \( C_2 \) and \( r \) is the radius of circle \( C_2 \).

### Step 6: Calculate the distance \( d \)
The distance \( d \) from point \( P(2, 0) \) to the center of circle \( C_2(-2, 0) \) is:
\[ d = |2 - (-2)| = 4 \]

### Step 7: Calculate the length of the tangent
The radius \( r \) of circle \( C_2 \) is 2. Thus, the length of the tangent is:
\[ \text{Length of tangent} = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \]

### Step 8: Area of the quadrilateral
The area of the quadrilateral formed by the two tangents and the two radii is given by:
\[ \text{Area} = 2 \times \text{Length of tangent} \times \text{Radius} \]
Substituting the values:
\[ \text{Area} = 2 \times (2\sqrt{3}) \times 2 = 8\sqrt{3} \text{ square units} \]

Thus, the area of the quadrilateral is \( 8\sqrt{3} \) square units.