Question

If $z=-3+4i$ and $zw=-14+2i$ then

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

Answer 1

Answer: (A), (C)

The correct options are
A $\arg w=\frac{\pi }{4}$
C $|w|=2\sqrt{2}$

$zw=-14+2i$
$\because z=-3+4i$
$\therefore (-3+4i)w=-14+2i$
$\Rightarrow w=\frac{-14+2i}{-3+4i}$
$\Rightarrow w=\frac{(-14+2i)(-3-4i)}{(-3+4i)(-3-4i)}$
$\Rightarrow w=\frac{42+56i-6i+8}{9+16}$
$\Rightarrow w=\frac{50+50i}{25}$
$\Rightarrow w=2+2i$
$\Rightarrow |w|=\sqrt{2^2+2^2}=2\sqrt{2}$
$\tan \alpha =\left(\frac{|2|}{|2|}\right)=\frac{\pi }{4}$
As $w$ lies in first quadrant So,
$\arg w=\alpha =\frac{\pi }{4}$