The equation $|z-i|=|z-1|, i=\sqrt{-1}$, represents:

(B) the line through the origin with slope 1

(A) a circle of radius $\frac{1}{2}$

(D) the line through the origin with slope -1

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Complex Numbers

The equation

∣ z - i ∣ = ∣ z - 1∣, i = - 1

, represents:

Answer: (B)

Put

z = x + i y

, so

∣ z - i ∣ = ∣ z - 1∣

becomes

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

. Expanding,

x^{2} + y^{2} - 2 y + 1 = x^{2} - 2 x + 1 + y^{2}

, which collapses to

- 2 y = - 2 x

, i.e.

y = x

. Geometrically this is the perpendicular bisector of the segment joining

i

and

1

, the line through the origin with slope

1

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option