Question

if the no. (z-2/z+2) is purely imaginary , then find the value of |z|

Answer 1

z = x +iy    | z | = √ x² + y²

now z-2/z+2 = x+iy-2/x+iy +2  = (x-2) +iy/ (x+2) + iy

= [(x-2) + iy][(x+2) - iy] / [(x+2) + iy ] [(x+2) - iy]

= (x² -4)  + iy(x+2) - iy (x-2) - i²y²  /  (x+2)² - i²y²

= x² -4 +iy(x+2 - x + 2)  + y² / x² + 4 +4x + y²

=  x² + y² - 4  / x² +y² + 4x +4              + i (4y) / x² +y² + 4x + 4

purely imaginary      Re = 0

x² + y² -4 = 0

⇒ x²+y² = 4  ====  √x² + y² = 2


hope this helps.