Question

Where does z lie, if ${\displaystyle \left|\frac{z-5i}{z+5i}\right|}$ = 1.

Answer 1

Given that: ${\displaystyle \left|\frac{z-5i}{z+5i}\right|}$ = 1

Let z = x + yi

∴ ${\displaystyle \left|\frac{x+yi-5i}{x+yi+5i}\right|}$ = 1

⇒ ${\displaystyle \left|\frac{x+(y-5)i}{x+(y+5)i}\right|}$ = 1

⇒ ${\displaystyle |x+(y-5)i|=|x+(y+5)i|}$

⇒ ${\displaystyle x^2+{(y-5)}^2=x^2+{(y+5)}^2}$

⇒ ${\displaystyle {(y-5)}^2={(y+5)}^2}$

⇒ y² + 25 – 10y = y² + 25 + 10y

⇒ 20y = 0

⇒ y = 0

Hence, z lies on x-axis i.e., real axis.