Confused by the question

[phalanx123]

Hi I was asked to do a question on Complex numbers. Here is the question

Describe fully the curve in the Argand diagram whose equation is
|z + 1 + i| = 8 .

Describe fully the three loci determined, as z moves round this curve, by the three complex numbers u, v and w defined as follows:
(i) u = 2x + iy (where z = x + iy );
(ii) v = z + 4 + 3i ;
(iii) w = iv .
I got the first part no problems, its the second part - what does it mean by "as z moves round this curve, by the three complex numbers u, v and w" :confused: I am really confused. Does it mean replace the z in the equation |z + 1 + i| = 8 by u,v and w each time or something else? could somebody help me. Thanks a million.:frown:

[AKG]

Regard u as a function that maps x + iy to 2x + iy.
Regard v as a function that maps x + iy to (x+4) + (3+y)i
Regard w as a function that maps x + iy to (-3-y) + (x+4)i

Find the image of the curve {z : |z+1+i| = 8} under the "functions" u, v, and w.

[phalanx123]

Ok, does it mean, taking the function v for example. the locus would be given by
|(x+4) + (3+y)i+1+i| = 8
which is |(x+5)+(4+y)i|=8

where z in |z+1+i| = 8 is replaced by the new "function" v=(x+4) + (3+y)i? Thanks

[AKG]

No. Draw the curve defined by |z + 1 + i| = 8 using black ink, let's call this curve (it's a circle) C. Pick a point somewhere on C, let's call this point x0 + iy0. Draw a red dot at the point 2x0 + iy0. Pick another point x1 + iy1 on C. Draw a red dot at the point 2x1 + iy1. Do this for every point on C. The red curve is the locus determined by v as z moves around C. Describe this curve, and you're done question (i).