Geometric interpretation of $|\frac{z+i} {z-i}| =2$

From $|\frac{z+i} {z-i}| = 2$, we get $\frac{|z – (–i)|} {|z – (i)|} = 2$.

If we let P, A, B to represent the complex numbers z, +i, and -i respectively, we have $\frac{BP} {AP} = \frac 21$.

enter image description here

This means we have another point C lying on AB such that PC is the angle bisector of $\angle APB$. For details, see the “angle bisector theorem”.

The same is true for the existence of another point D such that PD is the external angle bisector of $\angle APB$.

Note that (1) the angle between the internal and external angle bisector of the same angle is $\frac {\pi}{2}$; and (2) C, D are fixed points on AB. Hence, P lies on the circle with CD as diameter.


Here is a sketch of what geometrically is going on:

enter image description here

NOTE

As AB is constant, the equation describes a circle known as the Circle of Apollonius.

https://en.wikipedia.org/wiki/Circles_of_Apollonius

Tags:

Geometry

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