Let S = S 1 S 2 S 3 , where S 1 = { z : | z | 4 } , S 2 = { z: Im [ z - 1 + 3 ? i 1 - 3 ? i ] 0 } , and S 3 = { z : Re ( z ) 0 } . m i n z S | 1 - 3 i - z | = [2013]

Question

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. $\underset{z \in S}{m i n}$ $| 1 - 3 i - z |$ $=$

Smart Achievers · Accepted Answer

(3)

$S_{1} : x^{2} + y^{2} < 16$

$S_{2} : Im [\frac{(x - 1) + i (y + \sqrt{3})}{1 - i \sqrt{3}}] > 0$

$\Rightarrow \sqrt{3} (x - 1) + (y + \sqrt{3}) > 0 \Rightarrow y + \sqrt{3} x > 0$

$S_{3} : x > 0$

$Then S : S_{1} \cap S_{2} \cap S_{3} is as shown in the figure given below.$

$\min_{z \in S} | 1 - 3 i - z | = minimum distance between z and (1, - 3)$

$Clearly (from figure) minimum distance between z \in S and (1, - 3)$

$from line y + x \sqrt{3} = 0 i.e.$ $| \frac{\sqrt{3} - 3}{\sqrt{3} + 1} |$ $= \frac{3 - \sqrt{3}}{2}$

Solution

Q.
Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. $\underset{z \in S}{m i n}$ $| 1 - 3 i - z |$ $=$

1 $\frac{2 - \sqrt{3}}{2}$

2 $\frac{2 + \sqrt{3}}{2}$

3 $\frac{3 - \sqrt{3}}{2}$

4 $\frac{3 + \sqrt{3}}{2}$

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Solution

Q. Let S=S1∩S2∩S3, where S1={z∈ℂ:|z|<4}, S2={z∈ℂ:Im[z-1+3 i1-3 i]>0}, and S3={z∈ℂ:Re(z)>0}. [2013] Q. minz∈S|1-3i-z|=

1 2-32

2 2+32

3 3-32

4 3+32