Question

If arg $\left(\frac{z - \left(10 + 6 i\right)}{z - \left(4 + 2 i\right)}\right)=\frac{\pi }{4}$ (where $z$ is a complex number), then the perimeter of the locus of $z$ is

• A. $\frac{\sqrt{13} \pi }{4}$ units
• B. $\frac{3 \sqrt{13} \pi }{4}$ units
• C. $3\sqrt{13}\pi $ units
• D. $\frac{3 \pi }{2}\sqrt{26}$ units

Answer 1

Answer: (D)

Let $A=\left(4,2\right)$ , $B=\left(10,6\right)$ and $C$ is the center of locus of $z$ (which is a circle)

$\Rightarrow CA=CB=radius$ and $\angle ACB=2\left(\frac{\pi }{4}\right)=\frac{\pi }{2}$
$\Rightarrow \Delta ACB$ is right-angled isosceles triangle
$\Rightarrow r^{2}+r^{2}=\left(10 - 4\right)^{2}+\left(6 - 2\right)^{2}=36+16=52$
$\Rightarrow r^{2}=26\Rightarrow r=\sqrt{26}$
$\Rightarrow $ perimeter $=\frac{3}{4}\left(2 \pi r\right)$
$=\frac{3 \pi }{2}\sqrt{26}$ units