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  • The number of elements in the set  <img alt="\mathrm{\{z=a+i b \in \mathbb{C}: a, b \in \mathbb{Z}\; and\; \mathrm{1<|z-3+2 i|<4\}}}" src="https://entrancecorner.oncodecogs.com/gi

The number of elements in the set
 \mathrm{\{z=a+i b \in \mathbb{C}: a, b \in \mathbb{Z}\; and\; \mathrm{1<|z-3+2 i|<4\}}} is

Option: 1

40


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

Let us shift the origin to \mathrm{\left ( 3,-2 \right )}
So  \mathrm{1< \left | \mathbb{Z} \right |< 4}
No. of integral coordinates in the given region will be the same as in \mathrm{1< \left | \mathbb{Z}-3+2i \right |< 4}

No.of such point in the first quadrat are
\mathrm{\left ( 1,1 \right ),\left ( 2,1 \right ),\left ( 3,1 \right ),\left ( 1,2 \right )\left ( 2,2 \right ),\left ( 3,2 \right ),\left ( 1,3 \right ),\left ( 2,3 \right )= 8}
So in II, III & IV quadrant = 8,8,8.
On + x-axis : \mathrm{\left ( 2,0 \right )\&\left ( 3,0 \right )= 2}
On -ve x-axis = 2
On +ve y-axis = 2
On -ve y-axis = 2

Total = \mathrm{8+8+8+8+2+2+2+2= 40.}
 

Posted by

Suraj Bhandari

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