Answer to Question #2831 in Complex Analysis for Waqas

Question

Find the locus of points in the plane satisfying each of the given conditions:

(i) |z-5| = 6
(ii) |z-2i| >=1
(iii) Re(z+2) = -1

(iv) Re{i(conjugate of z)} =3
(v) |z+i| = |z-i|

Also Sketch its diagram.
Please tell me as soon as possible...?

Answer 1

(i) |z-5| = 6
This is a circle of radius 6 with center at 5

(ii) |z-2i| >=1
This is a outer part of a circle of radius 1 with center at 2i together with this circle

(iii) Re(z+2) = -1
This is a vertical line x+2=-1, that is x=3.

(iv) Re{i(conjugate of z)} =3
Let z=x+iy, then
i(conjugate of z) = ix-y,
whence
3 = Re{i(conjugate of z)} =
= Re{-y+ix}= -y
Thus we obtain a horizontal line y=-3.


(v) |z+i| = |z-i|
or any r the set
& |z+i|=r and |z-i|=r
Are two circles of radius r centered at -i and I respectively.
Then |z+i| = |z-i| is the set of all points equidistant to i and –i
This is a real axis y=0.