Question

If the maximum possible principal argument of the complex number $z$ satisfying $|z-4|=Re\left(z\right)$ is​​​​​​ $k$, then the value of $\frac{\pi }{k}$ is

Answer 1

Let $z=x+iy$
$|z-4|=Re\left(z\right)\Rightarrow \sqrt{(x-4{)}^2+y^2}=x\Rightarrow x^2-8x+16+y^2=x^2\Rightarrow y^2=8(x-2)$
This represents a parabola whose directrix is $x=0$


$\arg \left(z\right)$ is maximum possible when
The line drawn from origin is tangent to the parabola
As $x=0$ is directrix, so angle between the pair of tangents is $\frac{\pi }{2}$

Therefore, the maximum possible value is
$\arg \left(z\right)=\frac{\pi }{4}\therefore \frac{\pi }{k}=4$