Discussion
The area of the region bounded by \(\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1\) and the line segment joining (0,5) and (4, 0) in the first quadrant is
Answer: E
The area is \(5\pi - 10\).
The region lies between the chord from \((0, 5)\) to \((4, 0)\) and the arc of the ellipse in the first quadrant:
\[ \begin{aligned} \text{quarter ellipse} &= \frac{\pi ab}{4} = \frac{\pi \times 4 \times 5}{4} = 5\pi \\ \text{triangle} &= \frac{1}{2}\times 4 \times 5 = 10 \end{aligned} \]
The required area is \(5\pi - 10\). \(10\pi - 5\) swaps the roles, and the options with \(4\pi\) use \(a = b = 4\).
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