Tangents are drawn from `(4,4)` to the circle `x ^(2) + y ^(2) - 2x -7 =0` to meet the circle at A and B. The area bounded by PA, PB and circle is

Options

A. `(27pi)/(4) `sq. units
B. `(27pi)/(2)` sq. units
C. `(23pi)/(4)` sq. units
D. none of these

Answer

To find the area bounded by the tangents from point P(4, 4) to the circle given by the equation \(x^2 + y^2 - 2x - 7 = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
The given circle equation is:
\[ x^2 + y^2 - 2x - 7 = 0 \]
We can rewrite it in standard form by completing the square for the \(x\) terms:
\[ (x^2 - 2x) + y^2 = 7 \]
Completing the square for \(x\):
\[ (x - 1)^2 - 1 + y^2 = 7 \implies (x - 1)^2 + y^2 = 8 \]
This shows that the center of the circle is \(C(1, 0)\) and the radius \(r = \sqrt{8} = 2\sqrt{2}\).

### Step 2: Find the Length of the Tangent from Point P
The length of the tangent from point \(P(4, 4)\) to the circle can be calculated using the formula:
\[ \text{Length of tangent} = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \]
where \((h, k)\) is the center of the circle and \(r\) is the radius. Here, \((h, k) = (1, 0)\) and \(r = 2\sqrt{2}\):
\[ \text{Length of tangent} = \sqrt{(4 - 1)^2 + (4 - 0)^2 - (2\sqrt{2})^2} \]
Calculating this:
\[ = \sqrt{3^2 + 4^2 - 8} = \sqrt{9 + 16 - 8} = \sqrt{17} \]

### Step 3: Find the Angle Subtended by the Tangents at Point P
Let \(\theta\) be the angle subtended by the tangents at point P. The relationship between the angle and the radius and length of the tangent is given by:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{r}{\text{Length of tangent}} = \frac{2\sqrt{2}}{\sqrt{17}} \]
Thus, we have:
\[ \theta = 2 \tan^{-1}\left(\frac{2\sqrt{2}}{\sqrt{17}}\right) \]

### Step 4: Calculate the Area of the Sector
The area of the sector formed by the tangents and the circle is:
\[ \text{Area of sector} = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{\theta r^2}{2} \]
Substituting \(r^2 = 8\):
\[ \text{Area of sector} = \frac{\theta \cdot 8}{2} = 4\theta \]

### Step 5: Calculate the Area of Triangle PAB
The area of triangle PAB can be calculated using:
\[ \text{Area of triangle} = \frac{1}{2} \cdot \text{base} \cdot \text{height} \]
The base is the length of the tangent (which is \(\sqrt{17}\)) and the height can be calculated using \(r \sin\left(\frac{\theta}{2}\right)\):
\[ \text{Area of triangle} = \frac{1}{2} \cdot \sqrt{17} \cdot (2\sqrt{2}) \cdot \sin\left(\frac{\theta}{2}\right) \]

### Step 6: Find the Bounded Area
The bounded area between the tangents and the circle is given by:
\[ \text{Bounded Area} = \text{Area of sector} - \text{Area of triangle} \]
Substituting the values we calculated:
\[ \text{Bounded Area} = 4\theta - \text{Area of triangle} \]

### Final Calculation
Now we can substitute the values of \(\theta\) and the area of the triangle to find the final answer.