The circle is given by \(x^2 + y^2 = 169\), which has a radius of \(\sqrt{169} = 13\). The line equation \(5x - y = 13\) intersects the circle, creating a segment.
The line intersects the circle at points \((5, 12)\) and \((0, -13)\), as shown in the solution diagram.
The area of the segment below the line is calculated by integrating from \(y = -13\) to \(y = 12\):
\[ \text{Area} = \int_{-13}^{12} \sqrt{169 - y^2} \, dy - \frac{1}{2} \times 25 \times 5 \]
After integrating, we get:
\[ \text{Area} = \frac{\pi}{2} \cdot \frac{169}{2} - \frac{65}{2} + \frac{169}{2} \sin^{-1} \frac{12}{13} \]
Comparing terms, we find \(\alpha = 169\) and \(\beta = 2\).
\[ \alpha + \beta = 169 + 2 = 171 \]
So, the correct answer is: 171
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32