Area inside the curve `x^2+y^2=169` and below the line `5x-y=13` is

Options

A. `(169pi)/4-169/2sin^(-1)(12/13)+65/2`
B. `(169pi)/4+169/2sin^(-1)(12/13)-65/2`
C. `(169pi)/4-169/2sin^(-1)(12/13)-65/2`
D. `(169pi)/4+169/2sin^(-1)(12/13)+65/2`

Answer

To find the area inside the curve \(x^2 + y^2 = 169\) (a circle with radius 13) and below the line \(5x - y = 13\), we can follow these steps:

### Step 1: Identify the equations
The equation of the circle is:
\[ x^2 + y^2 = 169 \]
This represents a circle centered at the origin (0,0) with a radius of 13.

The equation of the line can be rearranged to:
\[ y = 5x - 13 \]

### Step 2: Find the points of intersection
To find the points where the line intersects the circle, substitute \(y\) from the line's equation into the circle's equation:
\[ x^2 + (5x - 13)^2 = 169 \]
Expanding the equation:
\[ x^2 + (25x^2 - 130x + 169) = 169 \]
Combine like terms:
\[ 26x^2 - 130x + 169 - 169 = 0 \]
This simplifies to:
\[ 26x^2 - 130x = 0 \]
Factoring out \(26x\):
\[ 26x(x - 5) = 0 \]
Thus, the solutions are:
\[ x = 0 \quad \text{or} \quad x = 5 \]

### Step 3: Calculate corresponding \(y\) values
For \(x = 0\):
\[ y = 5(0) - 13 = -13 \]
For \(x = 5\):
\[ y = 5(5) - 13 = 12 \]
The points of intersection are \((0, -13)\) and \((5, 12)\).

### Step 4: Determine the area of the segment
The area we need to find is the area of the segment of the circle that is below the line. We can use the formula for the area of a segment:
\[ \text{Area of segment} = \frac{1}{2} r^2 (\theta - \sin \theta) \]
where \(r\) is the radius and \(\theta\) is the angle in radians.

### Step 5: Find the angle \(\theta\)
To find \(\theta\), we first need to calculate \(\alpha\), the angle formed by the radius to the point \((5, 12)\):
\[ \tan \alpha = \frac{12}{5} \]
Using the Pythagorean theorem, the hypotenuse (radius) is:
\[ r = \sqrt{5^2 + 12^2} = 13 \]
Thus,
\[ \sin \alpha = \frac{12}{13}, \quad \cos \alpha = \frac{5}{13} \]
The angle \(\theta\) is:
\[ \theta = \frac{\pi}{2} + \alpha \]

### Step 6: Substitute into the area formula
Now substituting into the area formula:
\[ \text{Area} = \frac{1}{2} \times 169 \left(\frac{\pi}{2} + \sin^{-1}\left(\frac{12}{13}\right) - \sin\left(\frac{\pi}{2} + \alpha\right)\right) \]
Since \(\sin\left(\frac{\pi}{2} + \alpha\right) = \cos \alpha = \frac{5}{13}\):
\[ \text{Area} = \frac{1}{2} \times 169 \left(\frac{\pi}{2} + \sin^{-1}\left(\frac{12}{13}\right) - \frac{5}{13}\right) \]

### Step 7: Final simplification
Calculating the area:
\[ \text{Area} = \frac{169\pi}{4} + \frac{169}{2} \sin^{-1}\left(\frac{12}{13}\right) - \frac{65}{2} \]

### Conclusion
The area inside the curve \(x^2 + y^2 = 169\) and below the line \(5x - y = 13\) is given by the expression derived above.