Question

The area (in square units) of the part of the circle \(x^2 + y^2 = 169\) which is below the line \( 5x - y = 13 \) is\[\frac{\pi \alpha}{2 \beta} - \frac{65}{2} + \frac{\alpha}{\beta} \sin^{-1} \left( \frac{12}{13} \right)\]where \( \alpha \) and \( \beta \) are coprime numbers. Then \( \alpha + \beta \) is equal to

Answer 1

Correct Answer: 171

Step 1: Identify the Circle and Line Equation

The provided circle equation is \(x^2 + y^2 = 169\), yielding a radius of \(\sqrt{169} = 13\). The line equation is \(5x - y = 13\). These define a segment of the circle.

Step 2: Determine Points of Intersection

The line intersects the circle at the points \((5, 12)\) and \((0, -13)\). These are illustrated in the accompanying diagram.

Step 3: Calculate the Area Below the Line

The area of the segment situated below the line is computed via integration from \(y = -13\) to \(y = 12\):

\[ \text{Area} = \int_{-13}^{12} \sqrt{169 - y^2} \, dy - \frac{1}{2} \times 25 \times 5 \]

Step 4: Simplify the Result

The integrated result is:

\[ \text{Area} = \frac{\pi}{2} \cdot \frac{169}{2} - \frac{65}{2} + \frac{169}{2} \sin^{-1} \frac{12}{13} \]

Step 5: Determine \(\alpha\) and \(\beta\)

By comparing the simplified area expression, we identify \(\alpha = 169\) and \(\beta = 2\).

Step 6: Calculate \(\alpha + \beta\)

\[ \alpha + \beta = 169 + 2 = 171 \]

The final answer is: 171