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
If the four distinct points $ (4, 6) $, $ (-1, 5) $, $ (0, 0) $ and $ (k, 3k) $ lie on a circle of radius $ r $, then $ 10k + r^2 $ is equal to
The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:
Let the equation $ x(x+2) * (12-k) = 2 $ have equal roots. The distance of the point $ \left(k, \frac{k}{2}\right) $ from the line $ 3x + 4y + 5 = 0 $ is
Match List-I with List-II: List-I