how does one find the locus of a set of complex numbers defined in the form |z-a|=k|z-b|
for example in the question (CIE ALEVELS MATHS/9709/May-June 2013/Paper 33/Question 7) below we have to find the locus of the set of complex numbers |z-10i|=2|z-4i|
i looked through the internet and got to know that such equation is not represented by an angular bisector in the argan diagram as i previous thought but by a circle
an answer with respect to the question would be really appreciated (especially part iii)
Distance from a point is $k$ times the distance from another point where $k\neq1,>0$, the locus is a circle. It is called Apollonius circle.
You can prove it simply putting $z=x+i y$ and squaring both sides. You can find the centre and radius by locating $2$ points of circle on the line joining both points using section formula. These two will be diametrically opposite points.