Question

If $z$ is a complex number, then $|3z-1|=3|z-2|$ represents

• A. $y$-axis
• B. a circle
• C. $x$-axis
• D. a line parallel to $y$-axis

Answer 1

Answer: (D)

Let $z=x+iy$, then
$\left|3z-1\right|=3\left|z-2\right|$
$\Rightarrow \left|3\left(x+iy\right)-1\right|=3\left|x+iy-2\right|$
$\Rightarrow \left|\left(3x-1\right)+3iy\right|=3\left|x-2+iy\right|$
$\Rightarrow {\left(3x-1\right)}^2+9y^2=9\left[{\left(x-2\right)}^2+y^2\right]$
$\Rightarrow 9x^2+1-6x+9y^2$
$=9x^2+36-36x+9y^2$
$\Rightarrow 30x=35$
$\Rightarrow x=\frac{7}{6}$.
i.e., a st. line $\left|\right|$ to $y$-axis.