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 } . Area of S = [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. Area of $S =$

Smart Achievers · Accepted Answer

(2)

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

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

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

$S_{3} : x > 0$

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

Area of shaded region

$= \frac{π}{4} \times 4^{2} + \frac{π \times 4^{2} \times 60 °}{360 °}$

$= 4 π + \frac{8 π}{3} = \frac{20 π}{3}$

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. Area of $S =$

1 $\frac{10 π}{3}$

2 $\frac{20 π}{3}$

3 $\frac{16 π}{3}$

4 $\frac{32 π}{3}$

Want Us to Email you About Special Offers & Updates?

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. Area of S=

1 10π3

2 20π3

3 16π3

4 32π3