Normally I'd put more depth into my answer but I'm extremely busy so I'm only going to provide a sketch solution.

Firstly, ensure you understand the locus of \( \arg \left( \frac{z-z_1}{z-z_2} \right) = \theta \) when \(\theta\) is NOT equal to \(0\) or \(\pi\).

You can use this video by Eddie Woo to start thinking about it.After you've understood it, you'll note that \(\theta\) is the angle that the two line segments make

**on the arc of the circle**. What happens as \(\theta\) approaches \(\pi\)? The answer is that the arc of the circle

**gradually flattens out**. We know that when \(\theta=\frac\pi2\) we have a semi-circle, and when \(\frac\pi2 < \theta < \pi\) we have a minor-arc. That minor arc has to be stretched further and further out if we keep increasing the angle.

And then what happens when \(\theta = \pi\), i.e. it actually reaches \(\pi\)? Well the two line segments merge into one line segment to form the straight angle. The only way that can happen (in an intuitive sense) is if the arc flattens out into becoming a straight line segment as well.

On the contrary, when \( 0 < \theta < \frac\pi2\) we know that we have a major arc. What happens as \(\theta\) constantly decreases, and also when \(\theta\to 0\)? That angle being made is becoming smaller and smaller, and that sorta implies that the two line segments are getting closer and closer together. The only way this is possible is if that major arc is being inflated ever so large so that we can actually incorporate such a small angle.

So what do we 'intuitively' expect when \(\theta = 0\). That major arc is going to be inflated so much that its limiting behaviour is that essentially it's been popped open. Leaving you with two diametrically opposed rays.

(Note: Why should they be diametrically opposed and not pointing in random angles from each other? This is harder to describe intuitively and relies on you understanding what the limiting gradient-of-the-tangent of the arc at the two endpoints should be.)