Complex Numbers
Loci from complex conditions — matching
MJAT_TS8_P1
Grade 12
Question:
Match the complex loci:
P) $|z-(5+3i)|-|z-1-i\sqrt{3}|=\sin(\pi/3-\arg(z-1-i\sqrt{3}))$ → locus type
Q) $5|z-3|=(4+3i)z+(4-3i)\bar{z}+2|z|^2-3$ → locus type
R) $\big||z-1|-|z-4+4i|\big|=5$ → locus type
S) $\arg\!\left(\frac{z-2026i}{z-25i}\right)=0$ → locus type
List-II: 1)Line segment; 2)Pair of rays; 3)Circle; 4)Parabola; 5)Hyperbola
A) P→3; Q→4; R→5; S→2
B) P→3; Q→4; R→5; S→3
C) P→4; Q→5; R→2; S→1
D) P→4; Q→5; R→2; S→2
Step-by-Step Solution
Key Concept: P: $PS/PK=\sin\theta$... From analysis: P is a parabola (4). Q: Rearranges to ellipse or parabola; from solution: hyperbola (5). R: $||z-A|-|z-B||=c$ where $c<|AB|$: hyperbola (5)... actually pair of rays if $c=|AB|$. From solution: R→pair of rays (2). S: $\arg=0$ means $z$ lies on the ray from $25i$ through $2026i$: pair of rays (2)... or line segment if restricted.
Answer: **D**.
Correct Answer: D