Let `C: x ^(2) + y^(2)- 4x - 2y - 11=0 ` be a circle . A pair of tangents from the point (4,5) with a pair of radii form quadrilateral whose area is

Answer

To solve the problem, we need to find the area of the quadrilateral formed by the pair of tangents from the point (4,5) to the circle defined by the equation \( C: x^2 + y^2 - 4x - 2y - 11 = 0 \).

### Step-by-Step Solution:

1. **Rewrite the Circle Equation**:
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 4x - 2y - 11 = 0 \]
We can complete the square for \( x \) and \( y \).

- For \( x^2 - 4x \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
- For \( y^2 - 2y \):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

Substituting back, we have:
\[ (x - 2)^2 - 4 + (y - 1)^2 - 1 - 11 = 0 \]
Simplifying gives:
\[ (x - 2)^2 + (y - 1)^2 - 16 = 0 \]
Thus, the equation of the circle in standard form is:
\[ (x - 2)^2 + (y - 1)^2 = 16 \]
This indicates that the center of the circle is \( (2, 1) \) and the radius \( r = 4 \).

2. **Find the Length of the Tangent**:
The formula for the length of the tangent \( L \) from a point \( (x_1, y_1) \) to the circle is given by:
\[ L = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \]
where \( (h, k) \) is the center of the circle and \( r \) is the radius.

Here, \( (x_1, y_1) = (4, 5) \), \( (h, k) = (2, 1) \), and \( r = 4 \):
\[ L = \sqrt{(4 - 2)^2 + (5 - 1)^2 - 4^2} \]
Calculating this:
\[ L = \sqrt{(2)^2 + (4)^2 - 16} = \sqrt{4 + 16 - 16} = \sqrt{4} = 2 \]

3. **Calculate the Area of the Quadrilateral**:
The area \( A \) of the quadrilateral formed by the pair of tangents and the radii is given by:
\[ A = L \times r \]
Substituting the values we found:
\[ A = 2 \times 4 = 8 \]

### Final Answer:
The area of the quadrilateral formed by the pair of tangents and the radii is \( 8 \).