Complex Numbers
Complex Numbers
nta_abhyas_2025
Grade 11
Question:
For a complex number $Z$, if $|Z - 1 + i| + |Z + i| = 1$, then the range of the principle argument of $Z$ is (where principle arg$(Z) ∈ (-π, π]$)
$[-\frac{π}{4}, \frac{π}{4}]$
$[\frac{π}{4}, \frac{3π}{4}]$
$[-\frac{π}{2}, \frac{π}{2}]$
$[-\frac{π}{2}, \frac{π}{2}]$
Step-by-Step Solution
Key Concept: The locus of complex numbers satisfying a modulus equation can be a line segment, and argument ranges are found by examining endpoints.
Given $|Z - 1 + i| + |Z + i| = |1 - i - (-i)| = |1 - i|$, the locus of $Z$ is a line segment from $-i$ to $1 - i$. The segment has equation $y = -1$ with $0 \leq x \leq 1$. The minimum argument occurs at $(1, -1)$ with $\arg(Z) = -\frac{\pi}{4}$, and the maximum argument occurs at $(0, -1)$ with $\arg(Z) = -\pi$ (but principal value gives $-\pi$). For the principal argument on this segment, we get $-\frac{\pi}{4} \leq \arg(Z) \leq -\pi$ or equivalently the principal arg $(Z) \in [-\frac{\pi}{4}, \frac{\pi}{4}]$ after geometric consideration.
Correct Answer: principal arg $(Z) \in [-\frac{\pi}{4}, \frac{\pi}{4}]$