Complex Numbers
Argument locus; arc of circle; perimeter
MJMT_Full_Test_11
Grade 12
Question:
If $\arg\!\left(\dfrac{z-(10+6i)}{z-(4+2i)}\right)=\dfrac{\pi}{4}$, then the perimeter of locus of $z$ is
$\sqrt{13}\cdot\dfrac{\pi}{4}$
$3\sqrt{13}\pi$
$3\sqrt{13}\cdot\dfrac{\pi}{4}$
$3\sqrt{26}\cdot\dfrac{\pi}{2}$
Step-by-Step Solution
Key Concept: Locus is an arc of a circle. The chord from $(4,2)$ to $(10,6)$ subtends angle $\pi/4$ at $z$. Radius $R=|PQ|/(2\sin\theta)=\sqrt{52}/(2\sin(\pi/4))=\sqrt{52}/\sqrt2=\sqrt{26}$. Arc angle $=\pi-\pi/2=3\pi/2$... perimeter$=R\cdot\text{arc angle}=3\sqrt{26}\cdot\pi/2$.
Perimeter $=3\sqrt{26}\cdot\dfrac{\pi}{2}$.
Correct Answer: 4