Question:
The equation \(|z - i| = |z - 1|\), \(i = \sqrt{-1}\), represents :
The equation \(|z - i| = |z - 1|\), \(i = \sqrt{-1}\), represents :
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\(|z-i|=|z-1|\) means z is equidistant from the points \(i\), i.e. (0,1), and 1, i.e. (1,0): the perpendicular bisector of the segment joining them.
Updated On: Nov 5, 2026
- a circle of radius \(1\)
- the line through the origin with slope \(-1\)
- a circle of radius \(\frac{1}{2}\)
- the line through the origin with slope \(1\)
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The Correct Option is D
Solution and Explanation
Let \(z=x+iy\). Then \(|z-i|^2=x^2+(y-1)^2\) and \(|z-1|^2=(x-1)^2+y^2\).
Equating: \(x^2+y^2-2y+1=x^2-2x+1+y^2\Rightarrow -2y=-2x\Rightarrow y=x\).
This is a straight line through the origin with slope 1.
Answer: (4) the line through the origin with slope \(1\)
Equating: \(x^2+y^2-2y+1=x^2-2x+1+y^2\Rightarrow -2y=-2x\Rightarrow y=x\).
This is a straight line through the origin with slope 1.
Answer: (4) the line through the origin with slope \(1\)
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