The area (in sq. units) of the region \(\left\{(x, y): \displaystyle\frac{x^{2}}{9}+\frac{y^{2}}{4}\le 1,\ y \ge 2x\right\}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(3\pi\)
The region is the part of the ellipse on or above the line y = 2x. This line passes through the origin, which is the centre of the ellipse.
If (x, y) is inside the ellipse, so is (−x, −y), because the equation has only \(x^{2}\) and \(y^{2}\). If a point (x, y) has y > 2x, then for the point (−x, −y) the y-coordinate −y is less than 2(−x), so that point lies below the line. So the parts of the ellipse on the two sides of the line are equal, and the line cuts the ellipse into two halves.
The ellipse has semi-axes 3 and 2:
\( \displaystyle A = \frac{1}{2}\times\pi\times3\times2 = 3\pi \)