Question

The complex number $x+iy$ whose modulus is unity, $y\ne 0$, can be represented as $x+iy=\frac{a+i}{a-i}$, where $a$ is real number.

• A. True
• B. False

Answer 1

The correct option is A True

$\frac{a+i}{a-i}$$=\frac{(a+i{)}^2}{a^2+1}=\frac{a^2-1+2ai}{a^2+1}$

$\frac{a^2-1}{a^2+1}+\frac{2ai}{a^2+1}=z$(Let)

$|z|=\sqrt{\frac{(a^2-1{)}^2}{(a^2+1{)}^2}+(\frac{2a}{a^2+1}{)}^2}$

$⟹\frac{\sqrt{(a^2+1{)}^2}}{a^2+1}=1$

Hence true.