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

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> 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      Log On


   

Question 1172510: If z = %28x%2Bi%28y%2B2%29%29%2F%28%28x%2B1%29%2Biy%29, w = x+iy and |z| = 2,
Find the locus of w.

The answer is
%28x+%2B+4%2F3%29%5E2+%2B%28y-+2%2F3%29%5E2+=+20%2F9 , but I'm having some complications with the working
Answer by ikleyn(52798) About Me  (Show Source):
You can put this solution on YOUR website!
.

We are given that  abs%28%28x%2B+i%28y%2B2%29%29%2F%28%28x%2B1%29%2Biy%29%29 = 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


   3%2A%28x+%2B+%288%2F3%29x+%2B+16%2F9%29 + 3%2A%28y+-+%284%2F3%29y+%2B+4%2F9%29 = 16%2F3+%2B+4%2F3

   3%2A%28x+%2B+4%2F3%29%5E2 + 3%2A%28y+-+2%2F3%29%5E2 = 20%2F3

   %28x%2B4%2F3%29%5E2 + %28y-2%2F3%29%5E2 = 20%2F9.


It is EXACTLY the answer given in your post.


By the way, it is an equation of the circle of the radius of  sqrt%2820%29%2F3 = %282%2Asqrt%285%29%29%2F3  centered at the point  (-4%2F3,2%2F3).


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

Solved, answered, explained and completed.