Exercise 1570Algebra › Complex numbersOlympiadAdded 3 Nov 2026

A locus involving an angle

#locus#angle#complex

Statement

Describe the set of points $M(z)$ such that $\arg(z-1)\equiv\pi/4\pmod\pi$ .

Hint

$\arg(z-1)$ is the angle of the vector from $(1,0)$ to $M$ with the real axis.

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Solution

Interpretation. $z-1$ is the vector from $A(1,0)$ to $M$ . Its argument being $\pi/4$ modulo $\pi$ means $M$ lies on the line through $A$ with slope $\tan(\pi/4)=1$ .

Boundary. The line has equation $y=x-1$ , but $A$ itself is excluded (since $z-1=0$ has no argument).

Conclusion. The locus is the line $y=x-1$ with the point $A(1,0)$ removed.

A locus involving an angle

Statement

Solution

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