If denotes the number of solutions of |1-i|^x=2^x and =((|z|)/(arg(z))), where z=( )/(4)(1…

Question

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

Answer: 3

Accepted Answer

$\begin{gathered} (\sqrt{2}{)}^{x}={2}^{x}\Rightarrow x=0\Rightarrow \alpha =1 \ z=\frac{\pi }{4}(1+i{)}^{4}[\frac{\sqrt{\pi }-\pi i-i-\sqrt{\pi }}{\pi +1}+\frac{\sqrt{\pi }-i-\pi i-\sqrt{\pi }}{1+\pi }] \ =-\frac{\pi i}{2}(1+4i+6{i}^{2}+4{i}^{3}+1) \ =2\pi i \ \beta =\frac{2\pi }{\frac{\pi }{2}}=4 \end{gathered}$

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

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

Solutions · 1