Locus in the complex plane.

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

Originally Posted by jacks

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

What have you tried?

Originally Posted by jacks

Area of Region Bounded by the locus of $z$ which satisfy the equation $ \displaystyle \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 $ \displaystyle \arg(z + 5) - \arg(z - 5) = \pm \frac{\pi}{4}$, that is, $ \displaystyle \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 $ \displaystyle \arg(z +5) = \alpha$ and $ \displaystyle \arg(z - 5) = \beta$ subject to the restriction $ \displaystyle \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.