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)

$|w|=2\sqrt{2}$

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

As $w$ lies in first quadrant, so
$\arg w=\alpha =\frac{\pi }{4}$