Question

If $w=\frac{-9+3i}{1-2i}$, then

• A. $|w|=3\sqrt{2}$
• B. $\arg w=\frac{\pi }{4}$
• C. $|w|=\sqrt{2}$
• D. $\arg w=-\frac{3\pi }{4}$

Answer 1

Answer: (A), (D)

$\arg w=-\frac{3\pi }{4}$

$w=\frac{-9+3i}{1-2i}\Rightarrow w=\frac{(-9+3i)(1+2i)}{(1-2i)(1+2i)}\Rightarrow w=\frac{-9-18i+3i-6}{1+4}\Rightarrow w=\frac{-15-15i}{5}=-3-3i|w|=|-3-3i|=\sqrt{(-3{)}^2+(-3{)}^2}\therefore |w|=3\sqrt{2}$

Now,
$\tan \alpha =\left(\left|\frac{-3}{-3}\right|\right)=1$
$\Rightarrow \alpha =\frac{\pi }{4}$
As $w$ lies in third quadrant, so
$\arg w=-(\pi -\alpha )=-(\pi -\frac{\pi }{4})\therefore \arg w=-\frac{3\pi }{4}$