Locus Question - Complex Number

[frog0101]

Hi,
Could someone explain to me how to draw the locus of:


Thanks

[RuiAce]

These are essentially the two boundary cases of \( \arg \left( \frac{z-z_1}{z-z_2} \right)= \theta\). When \(\theta = 0\), what will happen is you get two pairs of rays pointed in opposite directions. When \(\theta = \pi\), what will happen is you get the line segment joining \(z_1\) and \(z_2\). (In a way, the cases \(\theta=0\) and \(\theta=\pi\) are opposing cases in this sense.) Both loci still exclude \(z_1\) and \(z_2\) as per usual.

An intuitive (but informal) explanation as to why this happens can be the following. As \(\theta \to \pi\), that angle \(z\) makes is becoming larger and larger. As that happens, the arc of the circle gets smaller, and eventually it flattens out into becoming a straight line.
Whereas as \(\theta \to 0\), that angle \(z\) makes is becoming smaller and smaller. As that happens, the arc of the circle keeps expanding at alarmingly faster rates. Eventually, when the circle has expanded in such a way that the radius is practically \(\infty\), it's degenerated into two lines, pointing in opposite directions from each other.