Question

Let $z=x+iy$ be a complex number such that ${\left(\frac{\overline{z}}{z}\right)}^i=e^{\varphi }$, where $\varphi ={\sin }^{-1}\left(\frac{24}{25}\right)$. If $x,y$ are natural numbers less than $10$, then the least possible value of $x+y$ is

Answer 1

Let $z=r(\cos \theta +i\sin \theta )=r{e}^{i\theta }$
${\left(\frac{\overline{z}}{z}\right)}^i=e^{\varphi }$
$\Rightarrow {\left(\frac{r{e}^{-i\theta }}{r{e}^{i\theta }}\right)}^i=e^{\varphi }$
$\Rightarrow (e^{-2i\theta }{)}^i=e^{\varphi }$
$\Rightarrow e^{2\theta }=e^{\varphi }$
$\Rightarrow 2\theta =\varphi ={\sin }^{-1}\left(\frac{24}{25}\right)$
$\Rightarrow \sin 2\theta =\frac{24}{25}$
$\Rightarrow \frac{2\tan \theta }{1+{\tan }^2\theta }=\frac{24}{25}\Rightarrow 25\tan \theta =12{\tan }^2\theta +12\Rightarrow 12{\tan }^2\theta -25\tan \theta +12=0\Rightarrow (3\tan \theta -4)(4\tan \theta -3)=0$

So, $\tan \theta =\frac{3}{4}$ or $\tan \theta =\frac{4}{3}$
Therefore, $\frac{y}{x}=\frac{3}{4}$ or $\frac{y}{x}=\frac{4}{3}$

Hence the minimum value of $x+y$ will be $7$.