The area (in sq. units) of the region S = { z ? C : | z - 1 | ? 2 ; ( z + z ¯ ) + i ( z - z ¯ ) ? 2 , Im ( z ) ? 0 } is, Complex Numbers jee mains questions | JEE Main Maths Complex Numbers Previous Year Question

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Solution

Q.
The area (in sq. units) of the region

$S = {z \in C : | z - 1 | \leq 2; (z + \bar{z}) + i (z - \bar{z}) \leq 2, Im (z) \geq 0}$ is [2024]

1 $\frac{7 π}{4}$

2 $\frac{3 π}{2}$

3 $\frac{7 π}{3}$

4 $\frac{17 π}{8}$

Ans.
(2)

$| z - 1 |$ $\leq 2 \Rightarrow {(x - 1)}^{2} + y^{2} \leq 4$

$z + \bar{z} + i (z - \bar{z}) \leq 2$

$\Rightarrow x + i y + x - i y + i (x + i y - x + i y) \leq 2$

$\Rightarrow x - y \leq 1$

$Im (z) \geq 0$

$\Rightarrow y \geq 0$

$∴$ $Required area = (\frac{\frac{3 π}{4}}{2 π})$ $(π) {(2)}^{2} = \frac{3}{2} π sq. units$

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