Question

If $\sqrt{5-12i}+\sqrt{-5-12i}=z$, then principal value of arg z can be

• A. $-\frac{\pi }{4}$
• B. $\frac{\pi }{4}$
• C. $\frac{3\pi }{4}$
• D. $-\frac{3\pi }{4}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $-\frac{\pi }{4}$
B $\frac{\pi }{4}$
C $\frac{3\pi }{4}$
D $-\frac{3\pi }{4}$

$5-12i=(3-2i{)}^2$
$-5-12i=(2-3i{)}^2$
$z=\sqrt{5-12i}+\sqrt{-5-12i}=\pm (3-2i{)}^{}\pm (2-3i{)}^{}$
Hence
$z=5-5i$ , arg $=\frac{-\pi }{4}$
$z=1+i$ , arg $=\frac{\pi }{4}$
$z=-1-i$ , arg $=\frac{-3\pi }{4}$
$z=-5+5i$, arg $=\frac{3\pi }{4}$
Hence, all options are correct.