The area (in sq. units) of the region enclosed by the ellipse \(4x^{2}+25y^{2}=100\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(10\pi\)
Dividing the equation by 100 gives the standard form of the ellipse:
\( \displaystyle \frac{x^{2}}{25}+\frac{y^{2}}{4}=1 \)
So the semi-axes are
\(a = 5\), \(b = 2\)
The ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) encloses an area \(\pi ab\):
\(A = \pi \times 5 \times 2 = 10\pi\)
Why the other options are incorrect
Option (1) \(25\pi\) — Incorrect. This is \(\pi a^{2}\), the area of the circle of radius 5; the ellipse is narrower, with b = 2.
Option (2) \(100\pi\) — Incorrect. This uses \(a^{2} = 25\) and \(b^{2} = 4\) in place of a and b: \(\pi \times 25 \times 4\).
Option (4) \(5\pi\) — Incorrect. This is the area of the upper half of the ellipse only.