Match List-I with List-II
| List-I (Region) | List-II (Area in sq. units) | ||
| (A) | Region enclosed by the circle \(x^{2}+y^{2}=7\) | (I) | 18 |
| (B) | Region enclosed by the ellipse \(\displaystyle\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\) | (II) | \(7\pi\) |
| (C) | Region enclosed by the curve \(|x|+|y|=3\) | (III) | 8 |
| (D) | Region bounded by the line x + y = 4 and the coordinate axes | (IV) | \(15\pi\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
(A) Region enclosed by the circle \(x^{2}+y^{2}=7\)
This matches (II) \(7\pi\).
A circle of radius r encloses the area \(\pi r^{2}\), and here \(r^{2} = 7\):
\(A = 7\pi\)
(B) Region enclosed by the ellipse \(\displaystyle\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
This matches (IV) \(15\pi\).
Here a = 3 and b = 5, and an ellipse encloses the area \(\pi ab\):
\(A = \pi\times3\times5 = 15\pi\)
(C) Region enclosed by the curve \(|x|+|y|=3\)
This matches (I) 18.
In the first quadrant the curve is the line x + y = 3, which cuts off a right triangle with legs 3 and 3. The curve is symmetric about both axes, so the region is four such triangles:
\( \displaystyle A = 4\times\frac{1}{2}\times3\times3 = 18 \)
(D) Region bounded by the line x + y = 4 and the coordinate axes
This matches (III) 8.
The line meets the axes at (4, 0) and (0, 4), and y = 4 − x:
\( \displaystyle A = \int_{0}^{4}(4-x)\,dx \)
\( \displaystyle = \left[4x-\frac{x^{2}}{2}\right]_{0}^{4} = 16-8 = 8 \)