An ellipse with centre at the origin has its major axis, of length 12, along the x-axis and its minor axis of length 8. The area (in sq. units) of the region enclosed by the ellipse is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(24\pi\)
The semi-axes are half the axis lengths:
\(a = 6,\ \ b = 4\)
so the ellipse is
\( \displaystyle \frac{x^{2}}{36}+\frac{y^{2}}{16} = 1 \)
The area enclosed by an ellipse is \(\pi ab\):
\(A = \pi\times6\times4 = 24\pi\)
Why the other options are incorrect
Option (1) \(96\pi\) — Incorrect. This uses the full lengths 12 and 8 in place of the semi-axes 6 and 4.
Option (2) \(48\pi\) — Incorrect. This is twice \(\pi ab\).
Option (3) \(6\pi\) — Incorrect. This is only the first-quadrant part, \(\frac{24\pi}{4}\).