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Q.
If $z$ is a complex number, then $ | 3z - 1 | = 3 | z - 2 |$ represents
Complex Numbers and Quadratic Equations
Solution:
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.