Question

The locus of $z$ given by $\left|\frac{z-1}{z-i}\right|=1$ is

• A. Circle
• B. Ellipse
• C. Straight line
• D. None of these

Answer 1

The correct option is C

Straight line

Explanation of the correct option.

Given: $\left|\frac{z-1}{z-i}\right|=1$

It can be written as , $\left|z-1\right|=\left|z-i\right|$

Put $z=x+iy$

$\left|x+iy-1\right|=\left|x+i\left(y-1\right)\right|$

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

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

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

$\Rightarrow$ $x=y$

Since it represents a straight line.

Hence option $\left(C\right)$ is the correct option.