Question

If $\alpha$ denotes the number of solutions of $|1 - i|^x = 2^x$ and $\beta = \frac{|z|}{\arg(z)}$, where $z = \frac{\pi}{4} (1 + i)^4 \left( \frac{1 - \sqrt{\pi} i}{\sqrt{\pi} + i} + \frac{\sqrt{\pi} - i}{1 + \sqrt{\pi} i} \right), \quad i = \sqrt{-1},$ then the distance of the point $(\alpha, \beta)$ from the line $4x - 3y = 7$ is _____

Answer 1

$(\sqrt{2})^x = 2^x \implies x = 0 \implies \alpha = 1$

$z = \frac{\pi}{4}(1 + i)^4$
$= \frac{-\pi i}{2} \left[ \sqrt{\frac{\pi - \pi i - \sqrt{\pi}}{\pi + 1}} + \sqrt{\frac{\pi - i - \pi i - \sqrt{\pi}}{1 + \pi}} \right]$
$= \frac{-\pi i}{2} \left( 1 + 4i + 6i^2 + 4i^3 + 1 \right)$
$= 2\pi i \beta = \frac{2\pi}{\pi/2} = 4$

Distance from $(1, 4)$ to $4x - 3y = 7$ will be:

$\frac{15}{5} = 3$