Two tangents to the circle `x^2 +y^2=4` at the points `A` and `B` meet at `P(-4,0)`, The area of the quadrilateral `PAOB`, where `O` is the origin, is

Options

A. 4
B. `6 sqrt(2)`
C. `4 sqrt(3)`
D. None of these

Answer

To find the area of the quadrilateral \( PAOB \), where \( P \) is the point where the tangents meet, \( A \) and \( B \) are the points of tangency on the circle, and \( O \) is the origin, we can follow these steps:

### Step 1: Understand the Circle
The equation of the circle is given by:
\[ x^2 + y^2 = 4 \]
This means the center of the circle \( O \) is at \( (0, 0) \) and the radius \( r \) is \( 2 \) (since \( r^2 = 4 \)).

**Hint:** Identify the center and radius of the circle from its equation.

### Step 2: Find the Coordinates of Points A and B
Since \( P(-4, 0) \) is outside the circle, we can find the points of tangency \( A \) and \( B \) using the property of tangents. The distance from the center \( O \) to point \( P \) is \( OP = 4 \). The length of the tangent from \( P \) to the circle can be calculated using the Pythagorean theorem:
\[ PA^2 = OP^2 - OA^2 \]
Where \( OA = 2 \) (the radius). Thus:
\[ PA^2 = 4^2 - 2^2 = 16 - 4 = 12 \implies PA = \sqrt{12} = 2\sqrt{3} \]

**Hint:** Use the Pythagorean theorem to find the length of the tangents.

### Step 3: Calculate the Angle of Tangents
The angle \( \theta \) between the radius and the tangent can be calculated using:
\[ \sin \theta = \frac{OA}{OP} = \frac{2}{4} = \frac{1}{2} \]
This means \( \theta = 30^\circ \).

**Hint:** Relate the lengths of the radius and the tangent to find the angle.

### Step 4: Calculate the Area of Triangle \( POA \)
The area of triangle \( POA \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, \( PO \) is the base and \( OA \) is the height. The area can also be calculated using:
\[ \text{Area} = \frac{1}{2} \times PA \times PO \times \sin \theta \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 2\sqrt{3} \times 4 \times \sin(30^\circ) = \frac{1}{2} \times 2\sqrt{3} \times 4 \times \frac{1}{2} = 2\sqrt{3} \]

**Hint:** Use the area formula for triangles and remember to incorporate the sine of the angle.

### Step 5: Calculate the Area of Quadrilateral \( PAOB \)
Since \( PAOB \) consists of two triangles \( POA \) and \( POB \), and both triangles are congruent:
\[ \text{Area of } PAOB = 2 \times \text{Area of } POA = 2 \times 2\sqrt{3} = 4\sqrt{3} \]

**Hint:** Recognize that the quadrilateral can be divided into two congruent triangles.

### Final Answer
Thus, the area of the quadrilateral \( PAOB \) is:
\[ \boxed{4\sqrt{3}} \]