Question

The complex number $3+4i$ is rotated about origin by an angle of $\frac{\pi }{4}$ and then stretched 2 times. The complex number corresponding to new position is

• A. $\sqrt{2}(-3+4i)$
• B. $\sqrt{2}(-1+7i)$
• C. $\sqrt{2}(3-4i)$
• D. $\sqrt{2}(1-7i)$

Answer 1

The correct option is C $\sqrt{2}(3-4i)$

$arg(3+4i)={\tan }^{-1}\left(\frac{4}{3}\right)$

New argument $=\frac{\pi }{4}+{\tan }^{-1}\left(\frac{4}{3}\right)={\tan }^{-1}\left(1\right)+{\tan }^{-1}\left(\frac{4}{3}\right)$

$={\tan }^{-1}\left(\frac{1+4/3}{1-4/3}\right)={\tan }^{-1}\left(\frac{4}{-1}\right)={\tan }^{-1}(-7)$

and if ${\left|z\right|}^2=k$

New ${\left|z\right|}^2=2k$

$\left|z\right|=\sqrt{2}\sqrt{k}$

New modulus is $\sqrt{2}$ times the old one. Therefore new complex number is

$\sqrt{2}(1-7i)$