JEE Mains · Maths · STD 11 - 4.1 complex nubers
The area (in sq. units) of the region \(\mathrm{S}=\{\mathrm{z} \in \mathbb{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{lm}(\mathrm{z}) \geq 0\}\) is
- A \(\frac{7 \pi}{3}\)
- B \(\frac{3 \pi}{2}\)
- C \(\frac{17 \pi}{8}\)
- D \(\frac{7 \pi}{4}\)
Answer & Solution
Correct Answer
(B) \(\frac{3 \pi}{2}\)
Step-by-step Solution
Detailed explanation
Put \(z=x+i y\) \( |z-1| \leq 2 \Rightarrow(x-1)^2+y^2 \leq 4 \) \( (z+\bar{z})+i(z-\bar{z}) \leq 2 \Rightarrow 2 x+i(2 i y) \leq 2 \) \( \Rightarrow x-y \leq 1 \) \( \operatorname{Im}(z) \geq 0 \Rightarrow y \geq 0\) Required area Area of semi-circle - area of sector A…
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