Question

If $|\sqrt{2} z-3+2 i|=|z|\left|\sin \left(\frac{\pi}{4}+\arg z_{1}\right)+\cos \left(\frac{3 \pi}{4}-\arg z_{1}\right)\right|$, 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)

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