Question

If $$-\pi <\arg \left(z\right)<-{\displaystyle \frac{\pi }{2}}$$, then $$\arg \left(\overline{z}\right)+\arg \left(-\overline{z}\right)$$ is equal to
A. $$\pi$$
B. $$-\pi$$
C. $${\displaystyle \frac{\pi }{2}}$$
D. $$-{\displaystyle \frac{\pi }{2}}$$

Answer 1

Hint: Let $$z=x+iy$$. Argument of the complex number $$z$$ is given. Using that identify the position of $$z$$. Then find its conjugate $$\overline{z}$$and negative of $$\overline{z}$$. After that find their arguments and add them to find the required answer.



Formula Used:If $$z=x+iy$$, then $$\overline{z}=x-iy$$
The point corresponding to a complex number lies in first quadrant if $$0<\arg \left(z\right)<{\displaystyle \frac{\pi }{2}}$$, lies in second quadrant if $${\displaystyle \frac{\pi }{2}}<\arg \left(z\right)<\pi$$, lies in third quadrant if $$-\pi <\arg \left(z\right)<-{\displaystyle \frac{\pi }{2}}$$, lies in fourth quadrant if $$-{\displaystyle \frac{\pi }{2}}<\arg \left(z\right)<0$$.



Complete step by step solution:Let $$z=x+iy$$, where $$x$$ and $$y$$ are real numbers and $$i=\sqrt{-1}$$
Given that $$-\pi <\arg \left(z\right)<-{\displaystyle \frac{\pi }{2}}$$
It means $$z$$ lies in third quadrant i.e. the values of both $$x$$ and $$y$$ are negative.
Let $$\arg \left(z\right)=-\pi +\theta$$, where $$0<\theta <{\displaystyle \frac{\pi }{2}}$$
Now, the conjugate of $$z$$ is $$\overline{z}=x-iy$$, which lies in second quadrant.
So, $$\arg \left(\overline{z}\right)=\pi -\theta$$
Again, $$-\overline{z}=-\left(x-iy\right)=-x+iy$$, which lies in first quadrant.
So, $$\arg \left(-\overline{z}\right)=\theta$$
Now, $$\arg \left(\overline{z}\right)+\arg \left(-\overline{z}\right)=\left(\pi -\theta \right)+\left(\theta \right)=\pi -\theta +\theta =\pi$$



Option ‘A’ is correct



Note: Argument of a complex number is the angle between the real axis and the line passes through origin and the point corresponding to the complex number. Many students can’t identify actual location of a complex number due to they can’t remember the ranges of the argument of the complex number quadrant-wise.