Exercise 835Algebra › Complex numbersL1Added 3 Nov 2026

The perpendicular bisector

#locus#perpendicular bisector#modulus

Statement

Describe the set of $z$ such that $|z-1|=|z+i|$ .

Hint

Points equidistant from $A(1,0)$ and $B(0,-1)$ .

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Solution

Interpretation. The condition means $MA=MB$ where $A(1,0)$ and $B(0,-1)$ . The locus is the perpendicular bisector of segment $[AB]$ .

Equation. The midpoint of $[AB]$ is $(1/2,-1/2)$ , and the direction of $AB$ is $(-1,-1)$ . The perpendicular bisector has direction $(1,-1)$ , so its equation is $y+1/2=-(x-1/2)$ , i.e. $x+y=0$ .

Conclusion. The locus is the line $y=-x$ .

The perpendicular bisector

Statement

Solution

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