Complex Numbers
Locus
Premium Question
Grade 11
Question:
The locus of $z$ satisfying $\arg\left(\frac{z - 1}{z + 1}\right) = \frac{\pi}{4}$ is:
(1) A straight line through $\pm 1$
(2) A full circle
(3) An arc of a circle
(4) A parabola
Step-by-Step Solution
Key Concept: The argument condition means the chord from $-1$ to $1$ subtends a fixed angle $\pi/4$ at $z$. By the inscribed angle theorem this is an arc of a specific circle passing through $1$ and $-1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)