Question

Let $z=x+iy$ where x, y are real variables and $i=\sqrt{-1}$. If $|2z-1|=|z-2|$, then the point z describes :

• A. A circle
• B. An ellipse
• C. A hyperbola
• D. A parabola

Answer 1

Answer: (A)

A circle

Given: $z=x+iy$

$\Rightarrow 2z-1=2x-1+2iy$ and $z-2=x-2+iy$

Given, $|2z-1|=|z-2|$

$\Rightarrow \sqrt{(2x-1{)}^2+(2y{)}^2}=\sqrt{(x-2{)}^2+y^2}$

$\Rightarrow (2x-1{)}^2+(2y{)}^2=(x-2{)}^2+y^2$

$\Rightarrow x^2+y^2=1$

This is a locus for circle, hence the point $z$ describes a circle.