If z-1/z+1 is purely imaginary number (z ≠ – 1), find the value of |z|.

Question

If \(\frac{z-1}{z+1}\) is purely imaginary number (z ≠ – 1), find the value of |z|.

Answer 1

Given: 

⇒ \(\frac{z-1}{z+1}\) is purely imaginary 

⇒ Let us assume \(\frac{z-1}{z+1}\) = ki, where K is any real number

Let us assume z=x+iy

⇒ \(\frac{x+iy-1}{x+iy+1}\) = ki

Multiplying and dividing with (x+1)-iy

Equating Real and Imaginary parts on both sides we get

⇒ \(\frac{x^2+y^2-1}{x^2+y^2+2x+1}\) 

⇒ x²+y²-1=0 

⇒ x²+y²=1

⇒\(\sqrt{ x^2+y^2}=\sqrt{1}\)

⇒ |z|=1 

∴ |z|=1