Area bounded by the circle which is concentric with the ellipse `(x^(2))/(25)+(y^(2))/(9) =1` and which passes through `(4,-(9)/(5))`, the vertical chord common to both circle and ellipse on the positive side of x-axis is

Options

A. `(481)/(25)tan^(-1)((9)/(20))-(36)/(5)`
B. `2 tan^(-1)((9)/(20))`
C. `(481)/(25)tan^(-1)((9)/(20))`
D. none of these

Explanation

To solve the problem of finding the area bounded by the circle that is concentric with the ellipse \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) and passes through the point \((4, -\frac{9}{5})\), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The given ellipse equation is:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
From this, we can identify:
- \(a^2 = 25 \Rightarrow a = 5\)
- \(b^2 = 9 \Rightarrow b = 3\)

### Step 2: Calculate the eccentricity of the ellipse
The eccentricity \(e\) of the ellipse can be calculated using the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values:
\[ e = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 3: Find the radius of the circle
Since the circle is concentric with the ellipse, its center is at the origin (0,0). The circle passes through the point \((4, -\frac{9}{5})\). We can calculate the radius \(r\) of the circle using the distance formula:
\[ r = \sqrt{(4 - 0)^2 + \left(-\frac{9}{5} - 0\right)^2} = \sqrt{16 + \frac{81}{25}} = \sqrt{\frac{400 + 81}{25}} = \sqrt{\frac{481}{25}} = \frac{\sqrt{481}}{5} \]

### Step 4: Determine the angle for the shaded area
The vertical chord common to both the circle and the ellipse can be determined by finding the angle \(\theta\) at which the line intersects the ellipse. Using the point \((4, -\frac{9}{5})\), we can find:
\[ \tan \theta = \frac{y}{x} = \frac{-\frac{9}{5}}{4} = -\frac{9}{20} \]
Thus, \(\theta = \tan^{-1}\left(-\frac{9}{20}\right)\).

### Step 5: Calculate the area of the shaded region
The area of the shaded region can be calculated using the formula for the area of the sector of the circle minus the area of the triangle formed by the points of intersection:
\[ \text{Area} = \left(\frac{2\theta}{360} \cdot \pi r^2\right) - \text{Area of triangle} \]
The area of the triangle can be calculated as:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times \left(-\frac{9}{5}\right) = -\frac{18}{5} \]

### Step 6: Substitute values and simplify
Substituting the values into the area formula:
1. Calculate the area of the sector:
\[ \frac{2\theta}{360} \cdot \pi \left(\frac{\sqrt{481}}{5}\right)^2 = \frac{2\theta}{360} \cdot \pi \cdot \frac{481}{25} \]
2. The area of the shaded region becomes:
\[ \text{Area} = \left(\frac{2\theta \cdot \pi \cdot 481}{900}\right) + \frac{18}{5} \]

### Final Answer
Thus, the area bounded by the circle and the ellipse is:
\[ \text{Area} = \frac{481}{25} \tan^{-1}\left(-\frac{9}{20}\right) - \frac{36}{5} \]