Complex Numbers 2 Question 1

1. The equation |z−i|=|z−1|,i=−1, represents

(a) a circle of radius 12 (2019 Main, 12 April I)

(b) the line passing through the origin with slope 1

(c) a circle of radius 1

(d) the line passing through the origin with slope -1

Show Answer

Answer:

Correct Answer: 1. (b)

Solution:

  1. Let the complex number z=x+iy

Also given, |z−i|=|z−1|

⇒|x+iy−i|=|x+iy−1|⇒x2+(y−1)2=(x−1)2+y2

[∵|z|=(Re(z))2+(Im(z))2]

On squaring both sides, we get

x2+y2−2y+1=x2+y2−2x+1

⇒y=x, which represents a line through the origin with slope 1.

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