If A$(z_1)$ and B$(z_2)$ are two fixed points in the Argand Plane and P$(z)$ is a variable point satisfying $|z-z_1| = k|z -z_2|$ then determine the locus of P
This is how I I proceeded with the problem :-
$|z-z_1| = k|z-z_2|$
Therefore ,this can be written as
$PA = kPB$
or we can say
$\frac{PA}{PB} = k$
I do not understand what to conclude from here . according to the solution of the problem , the previous equation indicates that the locus of P is a circle (or P lies on a circle) Now how does that make it a circle ? Please help
Actually, this is a circle unless $k=1$ (assuming that $k>0$).
Take $a,b,c,d\in\mathbb R$ such that $z_1=a+bi$ and $z_2=c+di$. Then, for each $x,y\in\mathbb R$,\begin{multline}|(x+yi)-(a+bi)|=k|(x+yi)-(c+di)|\iff\\\iff(x-a)^2+(y-b)^2=k^2\bigl((x-c)^2+(y-d)^2\bigr).\end{multline}Now, expand this equality and you'll get the equation of a circle (except if $k=1$, in which case you'll get a straight line).