If $\omega=\frac{z}{z-\frac{1}{3} i}$ and $|\omega|=1$, then $z$ lies on

JEE Main · 2005 · 08 May 2005 · Offline · Complex Numbers

ω = \frac{z}{z - \frac{1}{3} i}

and

∣ ω ∣ = 1

, then

z

lies on

(A)an ellipse
(B)a circle
(C)a straight line✓ Correct
(D)a parabola

Answer: (C)

Solution

From

∣ ω ∣ = 1

we have

\frac{∣ z ∣}{z - \frac{1}{3} i} = 1

, so

∣ z ∣ = z - \frac{1}{3} i

. This says that

z

is equidistant from the origin

(0, 0)

and the point

(0, \frac{1}{3})

. The set of points equidistant from two fixed points is the perpendicular bisector of the segment joining them, which is a straight line (here the horizontal line

y = \frac{1}{6}

). Hence

z

lies on a straight line.

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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