Complex numbers getting too complex! 
Let $z$ be on the curve $\arg\left(\dfrac{z-z_1}{z+z_1}\right)=\dfrac{\pi}{2}$. 
  If $\min\left(\arg\left(\dfrac{z}{z_1}\right),~\pi-\arg\left(\dfrac{z}{z_1}\right)\right)$ lies in $\left[\dfrac{\pi}{6},~\dfrac{\pi}{3}\right]$, then 
  $\left(\max\left(\lvert z-z_1\rvert\right)-\min\left(\lvert z+z_1\rvert\right)\right)$ is equal to 
  (A) $2\lvert z_1 \rvert$ 
  (B) $\left(\dfrac{\sqrt{2}-\sqrt{3}+1}{\sqrt{2}}\right)\lvert z_1 \rvert$ 
  (C) $\lvert z_1\rvert\sqrt{2}$ 
  (D) $\dfrac{\lvert z_1\rvert}{\sqrt{2}}$ 

As far as I reached, the curve on which $z$ lies is a circle centred at the origin with radius $\lvert z_1 \rvert$. The answer given is option (C). 
 A: As you've already found out, the complex number $z$ will lie on a semi-circle with centre at origin and radius $|z_1|$, like this:
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$
Now there is this condition:$$\min\left(\arg\left(\dfrac{z}{z_1}\right),~\pi-\arg\left(\dfrac{z}{z_1}\right)\right)\in\left[\dfrac{\pi}{6},~\dfrac{\pi}{3}\right]$$
First note that $\arg\left(\dfrac{z}{z_1}\right)$ is simply the angle between $z$, O and $z_1$ and  $\pi-\arg\left(\dfrac{z}{z_1}\right)$ is the angle between $z$, O and $z_2$
To use the condition given, bisect the semi-circle into two parts. In the part to the right, $\arg\left(\dfrac{z}{z_1}\right)$ will be lesser and in the other it will be greater than $\pi-\arg\left(\dfrac{z}{z_1}\right)$ as shown:
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$ 
In both the parts, it is specified that the corresponding angle should lie in the interval $\left[\dfrac{\pi}{6},\dfrac{\pi}{3}\right]$ so $z$ will be constrained to lie on the red arcs shown:
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$
It is quite clear from the diagram that $|z-z_1|$ will be maximum and $|z+z_1|$ will be minimum when $z$ is at the point A in above figure.
Maybe Brian Tung's hint will make more sense now.
