Question

The greatest positive argument of complex number satisfying $|z-4|=Re\left(z\right)$ is

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (D)

$\frac{\pi }{4}$

$|z-4|=Re\left(z\right)$
$\Rightarrow \sqrt{(x-4{)}^2+y^2}=x$
or $x^2-8x+16+y^2=x^2$
or $y^2=8(x-2)$
The given relation represents the part of the parabola with focus $(4,0)$ lying above the $x-axis$ and the imaginary axis as the directrix. The two tangents from directrix are at right angle. Hence, greatest positive argument of $z$ is $\frac{\pi }{4}$.