# If $\omega = \frac{z}{z-\left ( 1/3 \right )i}$ and $\left | \omega \right |= 1$,then $z$ lies on Option 1) a circle Option 2) an ellipse Option 3) a parabola Option 4) a straight line.

N neha
D Divya Saini

As we learnt in

Definition of Complex Number -

$z=x+iy, x,y\epsilon R$  & i2=-1

- wherein

Real part of z = Re (z) = x & Imaginary part of z = Im (z) = y

$\omega=\frac{z}{z-\frac{1}{3}i}$      &     $|\omega|=1$

Let z = x + iy

Now  $\omega=\frac{x+iy}{x+iy-\frac{i}{3}}=\frac{x+iy}{x+\left ( y-\frac{1}{3} \right )}$

$\therefore\ \; |\omega|^{2}=1^{2}=1=\frac{x^{2}+y^{2}}{x^{2}+\left(y-\frac{1}{3} \right )^{2}}$

$\therefore\ \; x^{2}+ \left(y-\frac{1}{3} \right )^{2}=x^{2}+y^{2}$

$\therefore\ \; y^{2}-\frac{2y}{3}+\frac{1}{9}=y^{2}$

$\therefore\ \; \frac{2y}{3}=\frac{1}{9}$

$\therefore\ \; y=\frac{1}{6}$    straight line parallel to x-axis.

Correct option is 4.

Option 1)

a circle

This is an incorrect option.

Option 2)

an ellipse

This is an incorrect option.

Option 3)

a parabola

This is an incorrect option.

Option 4)

a straight line.

This is the correct option.

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