Area Under Curves Question 5
Question 5 - 29 January - Shift 1
Let $A=\lbrace (x, y) \in \mathbb R^{2}: y \geq 0,2 x \leq y \leq \sqrt{ 4-(x-1)^{2}}\rbrace$ and $B=\lbrace (x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min {2 x, \sqrt{4-(x-1)^{2}}}\rbrace$
Then the ratio of the area of A to the area of B is
(1) $\frac{\pi-1}{\pi+1}$
(2) $\frac{\pi}{\pi-1}$
(3) $\frac{\pi}{\pi+1}$
(4) $\frac{\pi+1}{\pi-1}$
Show Answer
Answer: (1)
Solution:
Formula: Area between two curves - Area enclosed between two curves intersecting at two different points
$y^{2}+(x-1)^{2}=4$
Area of shaded portion $=circular(OABC) - Ar(\triangle OAB)$
Area of shaded portion $=\frac{\pi(4)}{4}-\frac{1}{2}(2)(1)$
$A=(\pi-1)$
Area $B=Ar(\triangle AOB)+$ Area of arc of circle $(ABC)$
Area $=\frac{1}{2}(1)(2)+\frac{\pi(2)^{2}}{4}=\pi+1$
Area $\frac{A}{B}=\frac{\pi-1}{\pi+1}$
So, the correct option is (1)
