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Help me answer: - Complex numbers and quadratic equations - JEE Main-2

The line joining origin and the point represented by z=1+i  is rotated through an angle \frac{3\pi }{2} in anticlockwise direction about origin and stretched by additional \sqrt{2} units. In the new position , point is represented by the complex number

  • Option 1)

    -\sqrt{2}-\sqrt{2}i

  • Option 2)

    \sqrt{2}-\sqrt{2}i

  • Option 3)

    2-\sqrt{2}i

  • Option 4)

    2-2i

 
Answers (1)
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using rotation at 0 

\frac{Z_{2}-0}{Z_{1}-0}=\frac{2\sqrt{2}}{\sqrt{2}}e^{i3\frac{\pi }{2}}

\Rightarrow \frac{Z_{2}}{Z_{1}}=2\left ( -i \right )\Rightarrow Z_{2}=-2i\left ( 1+i \right )=2-2i

 

Rotation -

\frac{z_{3}-z_{1}}{z_{2}-z_{1}}=\frac{\left |z_{3}-z_{1} \right |}{\left |z_{2}-z_{1} \right |}.e^{i\Theta }

 

 

- wherein

e^{i \theta}=\cos \theta+i\sin \theta

 

 


Option 1)

-\sqrt{2}-\sqrt{2}i

This is incorrect

Option 2)

\sqrt{2}-\sqrt{2}i

This is incorrect

Option 3)

2-\sqrt{2}i

This is incorrect

Option 4)

2-2i

This is correct

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