In each item of List-I, k is a positive constant. Match List-I with List-II
| List-I (Region and its area) | List-II (Value of k) | ||
| (A) | The region bounded by the line y = kx, the x-axis and the line x = 2 has area 6 | (I) | 2 |
| (B) | The region enclosed by the circle \(x^{2}+y^{2}=k\) has area \(4\pi\) | (II) | 3 |
| (C) | The region enclosed by the ellipse \(\displaystyle\frac{x^{2}}{k}+\frac{y^{2}}{4}=1\) has area \(6\pi\) | (III) | 4 |
| (D) | The region bounded by the curve \(y = kx^{2}\), the x-axis and the line x = 3 has area 18 | (IV) | 9 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
In each item the area is written in terms of k and then set equal to the given area.
(A) The region bounded by the line y = kx, the x-axis and the line x = 2 has area 6
This matches (II) 3.
\( \displaystyle \int_{0}^{2}kx\,dx = k\left[\frac{x^{2}}{2}\right]_{0}^{2} = 2k \)
\(2k = 6,\ \ k = 3\)
(B) The region enclosed by the circle \(x^{2}+y^{2}=k\) has area \(4\pi\)
This matches (III) 4.
The radius is \(\sqrt{k}\), so the area is \(\pi r^{2} = \pi k\):
\(\pi k = 4\pi,\ \ k = 4\)
(C) The region enclosed by the ellipse \(\displaystyle\frac{x^{2}}{k}+\frac{y^{2}}{4}=1\) has area \(6\pi\)
This matches (IV) 9.
The semi-axes are \(\sqrt{k}\) and 2, and the area is \(\pi ab\):
\(\pi\times\sqrt{k}\times2 = 6\pi\)
\(\sqrt{k} = 3,\ \ k = 9\)
(D) The region bounded by the curve \(y = kx^{2}\), the x-axis and the line x = 3 has area 18
This matches (I) 2.
\( \displaystyle \int_{0}^{3}kx^{2}\,dx = k\left[\frac{x^{3}}{3}\right]_{0}^{3} = 9k \)
\(9k = 18,\ \ k = 2\)