Question

A complex number $z=3+4i$ is rotated about another fixed complex number $z_1=1+2i$ in anticlockwise direction by ${45}^o$ angle. Find the complex number represented by new position of z in argand plane

• A. $1+(2-\sqrt{2})i$
• B. $1+(2+2\sqrt{2})i$
• C. $1-(2+2\sqrt{2})i$
• D. $1+(2+\sqrt{2})i$

Answer 1

Answer: (B)

$1+(2+2\sqrt{2})i$

A complex number when multiplied by $e^{i\theta }$ is rotated by $\theta$ in the anticlockwise direction.
Here, since the point about which the complex number has to be rotated is not origin, we need to subtract the fixed point from the given complex number.
So, the new z after rotation $=(3+4i-1-2i)\times e^{i\frac{\pi }{4}}+(1+2i)$
$\Rightarrow z=(2+2i)\times (\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})+(1+2i)$
$\Rightarrow z=2\sqrt{2}i+1+2i=1+i(2+2\sqrt{2})$