Q

# 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)

$2-2i$

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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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