The area (in sq. units) of the region inside the rectangle with vertices (4, 3), (−4, 3), (−4, −3) and (4, −3) but outside the ellipse \(\displaystyle\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(48-12\pi\)
The ellipse has semi-axes a = 4 and b = 3, so it touches the four sides of the rectangle at (±4, 0) and (0, ±3) and lies inside the rectangle. The required area is the rectangle area minus the ellipse area.
Rectangle: \(8\times6 = 48\)
The area enclosed by an ellipse is \(\pi ab\):
Ellipse: \(\pi\times4\times3 = 12\pi\)
\(A = 48-12\pi\)