The area (in sq. units) of the bigger portion of region enclosed by the curves \(4x^2+9y^2 = 36\) and \(2x + 3y = 6\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{3}{2} (3\pi+2)\)
Dividing the curve by 36 gives the ellipse
\(\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1\)
with a = 3 and b = 2. The line \(2x + 3y = 6\), that is \(\frac{x}{3} + \frac{y}{2} = 1\), meets the axes at (3, 0) and (0, 2), which are the ends of the semi-axes, so it is a chord of the ellipse that cuts it into two portions.
The smaller portion lies in the first quadrant, between the ellipse and the chord. It is the quarter ellipse minus the triangle with vertices (0, 0), (3, 0) and (0, 2):
\(\displaystyle \text{Quarter ellipse} = \frac{1}{4}\pi(3)(2) = \frac{3\pi}{2}\)
\(\displaystyle \text{Triangle} = \frac{1}{2}(3)(2) = 3\)
\(\displaystyle \text{Smaller portion} = \frac{3\pi}{2} - 3\)
The whole ellipse has area \(\pi ab = 6\pi\), so the bigger portion is
\(\displaystyle 6\pi - \left(\frac{3\pi}{2} - 3\right) = \frac{9\pi}{2} + 3\)
\(\displaystyle = \frac{3}{2}(3\pi + 2)\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{2} (\pi-2)\) — Incorrect. This is \(\frac{3\pi}{2} - 3\), the area of the smaller portion.
Option (3) \(6\pi\) — Incorrect. This is the area of the whole ellipse.
Option (4) \(3\pi+3\) — Incorrect. The bigger portion is \(\frac{9\pi}{2} + 3\), not \(3\pi + 3\).