Question

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

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

Answer 1

The correct option is B a circle

Let $z=x+iy$.

Given,

$|3z-1|=|z-2|$

or, $|(3x-1)+i\left(3y\right)|=|(x-2)+iy|$

or, $\sqrt{(3x-1{)}^2+(3y{)}^2}=\sqrt{(x-2{)}^2+(y{)}^2}$

Squaring both sides we get,

$9x^2-6x+1+9y^2=x^2-4x+4+y^2$

or, $8x^2+8y^2-2x-3=0$

or, $x^2+y^2-\frac{x}{4}-\frac{3}{8}=0$

This is an equation of a circle.