locus of a complex number in the form |z-a|=k|z-b|

Question

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)

the question

evil999man · Answer 1 · 2014-05-02T17:19:50.317

2

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.

edited May 02 '14 at 17:19

answered May 02 '14 at 17:14

evil999man

6,018

so what will be the center of the circle and how much will the radius be equal to? – phantom.omaga May 02 '14 at 17:18
@phantom.omaga edited. – evil999man May 02 '14 at 17:19
O.o section formula? sorry i am not getting it even after your explanation. Can you please explain it with reference to the question above or any other dummy values perhaps then it might go through my thick head :) sorry again. – phantom.omaga May 02 '14 at 17:28
Can you find centre and radius of circle by using its equation? It is an alternate method. – evil999man May 02 '14 at 17:30
and how is that done? – phantom.omaga May 02 '14 at 17:40
@phantom.omaga Google. – evil999man May 02 '14 at 17:45
FYI Google and Wikipedia no help so far. – phantom.omaga May 02 '14 at 17:52
@phantom.omaga Write equation as $(x-h)^2+(y-k)^2=r^2$ where (h,k) is centre and $r$ is radius. Expand this and compare with equation you got while putting z=x+iy . you will find h,k,r. – evil999man May 03 '14 at 04:30

locus of a complex number in the form |z-a|=k|z-b|

1 Answers1