Question

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

• A. $-\frac{\pi }{4}$
• B. $\pi$
• C. $2\pi$
• D. None

Answer 1

The correct option is A $-\frac{\pi }{4}$

$\sqrt{5-12i}=\sqrt{9+4i^2-2\cdot 3\cdot 2i}=\sqrt{(3-2i{)}^2}\Rightarrow \pm (3-2i)$

$\sqrt{-5-12i}=\sqrt{4+9i^2-2\cdot 2\cdot 3i}=\sqrt{(2-3i{)}^2}\Rightarrow \pm (2-3i)$

$z=\pm (3-2i)\pm (1-3i)$

$z=5-5i,1+i,-5+5i,-1-i$

principle values arg $\left(z\right)-3\pi /4,3\pi /4,\pi /4,-\pi /4$