Question

If $|\sqrt{2}z-3+2i|=|z|\left|\sin \left(\frac{\pi }{4}+\arg z_1\right)+\cos \left(\frac{3\pi }{4}-\arg z_1\right)\right|$, where $z_1=1+\frac{1}{\sqrt{3}}i$, then locus of $z$ is

• A. pair of straight lines
• B. circle
• C. parabola
• D. ellipse

Answer 1

Answer: (B)

$z_1=1+\frac{1}{\sqrt{3}}i$
$\therefore \arg \left(z_1\right)=\frac{\pi }{6}$
$\therefore \sin \left(\frac{\pi }{4}+\arg z_1\right)+\cos \left(\frac{3\pi }{4}-\arg z_1\right)=\frac{1}{\sqrt{2}}$
$\therefore |\sqrt{2}z-3+2i|=|z|\frac{1}{\sqrt{2}}$
$\Rightarrow \left|\frac{z-\frac{3-2i}{\sqrt{2}}}{z}\right|=\frac{1}{2}$, which represents a circle