Question

State true or false:

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.

Answer 1

Let the complex number $x+iy$ be $z$
$|z|=1$
Now 'a' is real number.
Then
$\frac{a-i}{a+i}=\frac{x}{\overline{x}}=\lambda$ where $x$ is complex number $x=a-i$
Thus
$|\lambda |=|\frac{x}{\overline{x}}|$
$=\frac{|x|}{|\overline{x}|}$
$=1$ ...(ii)
Hence $x+iy=\frac{a-i}{a+i}$ is possible and $|x+iy|=|\frac{a-i}{a+i}|=1$... from (ii).