The area (in sq. units) of the region inside the ellipse \(\displaystyle\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) and outside the circle \(x^{2}+y^{2}=9\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(6\pi\)
The ellipse has semi-axes a = 5 and b = 3. The circle has radius 3, equal to b, so it touches the ellipse at (0, ±3) and otherwise lies inside the ellipse. The required area is the ellipse area minus the circle area.
An ellipse encloses the area \(\pi ab\) and a circle the area \(\pi r^{2}\):
Ellipse: \(\pi\times5\times3 = 15\pi\)
Circle: \(\pi\times3^{2} = 9\pi\)
\(A = 15\pi-9\pi = 6\pi\)