Locus in the complex plane.

[juantheron]

Area of Region Bounded by the locus of $z$ which satisfy the equation $$\displaystyle \arg \left(\frac{z+5i}{z-5i}\right) = \pm \frac{\pi}{4}$$ is

 

[Mr Fantastic]

Area of Region Bounded by the locus of $z$ which satisfy the equation $$\displaystyle \arg \left(\frac{z+5i}{z-5i}\right) = \pm \frac{\pi}{4}$$ is

What have you tried?

 

[Mr Fantastic]

Area of Region Bounded by the locus of $z$ which satisfy the equation $$\displaystyle \arg \left(\frac{z+5i}{z-5i}\right) = \pm \frac{\pi}{4}$$ is

You can take a geometric approach.

Your relation can be written $$\arg(z + 5) - \arg(z - 5) = \pm \frac{\pi}{4}$$, that is, $$\alpha - \beta =\pm \frac{\pi}{4}$$.

Consider the line segment joining z = 5 and z = -5 as the chord on a circle and consider the rays $$\arg(z +5) = \alpha$$ and $$\arg(z - 5) = \beta$$ subject to the restriction $$\alpha - \beta =\pm \frac{\pi}{4}$$. Consider the intersection of these rays and the angle between them at their intersection point. The angle is constant ... Now think of a circle theorem involving angles subtended by the same arc at the circumference ...

It's not hard to see you that have a circle with 'holes' at z = 5 and z = -5 (why?).

Now your job is to determine the radius of this circle and use it to get the area.