Q. The locus of the complex number $z$ in the Argand plane if $∣ ∣ \frac{1 - i z}{z - i} ∣ ∣ = 1$ , is

1928 237 Complex Numbers and Quadratic Equations Report Error

A

a circle

B

$x$ -axis

C

$y$ -axis

D

None of these

Solution:

Let $z = x + i y$
Given, $∣ ∣ \frac{1 - i z}{z - i} ∣ ∣ = 1$
$\Rightarrow ∣ ∣ \frac{1 - i ( x + i y )}{x + i y - i} ∣ ∣ = 1$
$\Rightarrow ∣ ∣ \frac{1 + y - i x}{x + i ( y - 1 )} ∣ ∣ = 1$
$\Rightarrow \frac{( 1 + y ) ^{2} + x ^{2}}{x ^{2} + ( y - 1 ) ^{2}} = 1$
$\Rightarrow (1 + y)^{2} + x^{2} = x^{2} + (y - 1)^{2}$
$\Rightarrow 1 + y^{2} + 2 y + x^{2} = x^{2} + y^{2} - 2 y + 1$
$\Rightarrow 4 y = 0$
$\Rightarrow y = 0$ ,
which is the equation of $x$ -axis.