Let C be the set of all complex numbers.

Question:

Let C be the set of all complex numbers. LetĀ 

$\mathrm{S}_{1}=\left\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-3-\left.2 \mathrm{i}\right|^{2}=8\right\}$

$\mathrm{S}_{2}=\{\mathrm{z} \in \mathrm{C} \mid \operatorname{Re}(\mathrm{z}) \geq 5\}$ and

$\mathrm{S}_{3}=\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-\overline{\mathrm{z}} \mid \geq 8\}$

Then the number of elements in $S_{1} \cap S_{2} \cap S_{3}$ is equal to

  1. 1

  2. 0

  3. 2

  4. InfiniteĀ 


Correct Option: 1

Solution:

$\mathrm{S}_{1}:|\mathrm{z}-3-2 \mathrm{i}|^{2}=8$

$|z-3-2 i|=2 \sqrt{2}$

$(x-3)^{2}+(y-2)^{2}=(2 \sqrt{2})^{2}$

$\mathrm{S}_{2}: \mathrm{x} \geq 5$

$\mathrm{S}_{3}:|\mathrm{z}-\overline{\mathrm{z}}| \geq 8$

$|2 \mathrm{iy}| \geq 8$

$2|\mathrm{y}| \geq 8 \quad \therefore \mathrm{y} \geq 4, \mathrm{y} \leq-4$

$\mathrm{n}\left(\mathrm{S}_{1} \cap \mathrm{S}_{2} \cap \mathrm{S}_{3}\right)=1$

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