Question

If arg $\left(\frac{z-\left(10+6i\right)}{z-\left(4+2i\right)}\right)=\frac{\pi }{4}$ (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