Question

If $z_1=-3+5i;z_2=-5-3i$ and $z$ is a complex number lying on the line segment joining $z_1$ and $z_2$, then $arg\left(z\right)$ can be

• A. $-\frac{3\pi }{4}$
• B. $-\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{5\pi }{6}$

Answer 1

The correct option is D $\frac{5\pi }{6}$

$arg\left(z\right)$ must be less than $arg\left(z_2\right)$ and greater than $arg\left(z_1\right)$.

$\because z$ is lying on the line segment joining $z_1$ and $z_2$.

$\therefore \arg \left(z_1\right)\le arg\left(z\right)\le arg\left(z_2\right)$

$\Rightarrow \frac{\pi }{2}+{\tan }^{-1}\frac{3}{5}\le arg\left(z\right)\le \pi +{\tan }^{-1}\frac{3}{5}$

we know, ${\tan }^{-1}\frac{3}{5}\approx \frac{\pi }{6}$

$\therefore \frac{2\pi }{3}\le arg\left(z\right)\le \frac{7\pi }{6}$

Thus option$D$ is correct.