SOLUTION: If z = {{{(x+i(y+2))/((x+1)+iy)}}}, w = x+iy and |z| = 2, Find the locus of w. The answer is {{{(x + 4/3)^2 +(y- 2/3)^2 = 20/9}}} , but I'm having some complications wi

Question 1172510: If z = , w = x+iy and |z| = 2,
Find the locus of w.

The answer is
, but I'm having some complications with the working
Answer by ikleyn(52798) (Show Source): You can put this solution on YOUR website!
.


We are given that   = 2.


It implies  | x + i(y+2) | = 2 * |(x+1) + iy |.


In turn, it means that

    x^2 + (y+2)^2 = 4 * ((x+1)^2 + y^2).


Simplify it a bit

    x^2 + y^2 + 4y + 4 = 4*(x^2 + 2x + 1 + y^2),

    x^2 + 4y + y^2 + 4 = 4x^2 + 8x + 4 + 4y^2,

    3x^2 + 8x + 3y^2 - 4y = 0.


Complete the squares separately for x-terms and y-terms


    +  = 

    +  = 

    +  = .


It is EXACTLY the answer given in your post.


By the way, it is an equation of the circle of the radius of   =   centered at the point  (,).


So, the locus of  { w }  is the described circle.     ANSWER

Solved, answered, explained and completed.

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