Complex Numbers
Locus from Distance Condition
nta_pyq_2023_jan
Grade None
Question:
Let $z_1=2+3i$ and $z_2=3+4i$. The set $S=\{z\in\mathbb{C}:|z-z_1|^2-|z-z_2|^2=|z_1-z_2|^2\}$ represents a:
straight line with sum of its intercepts on coordinate axes equals 14
hyperbola with the length of transverse axis 7
straight line with sum of its intercepts on coordinate axes equals -18
hyperbola with eccentricity 2
Step-by-Step Solution
Key Concept: Expand: $(x-2)^2+(y-3)^2-(x-3)^2-(y-4)^2=|(-1-i)|^2=2$. $2x+2y-12=2\Rightarrow x+y=7$.
Straight line $x+y=7$; sum of intercepts $=14$.
Correct Answer: 1