Question

Let $z_1=10+6i$ and $z_2=4+6i$ If z is any complex number such that the argument of $\left(\frac{z-z_1}{z-z_2}\right)$ is $\frac{\pi }{4}$ then prove that $|z-7-9i|=3\sqrt{2}$

Answer 1

Since, $z_1=10+6i,z_2=4+6i$
and arg $\left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4}$ represents locus of $z$ is a circle show
as from the figure whose centre is $(7,y)$ and $\angle AOB={90}^0$ clearly, $OC=9$
$\Rightarrow OD=6+3=9$
$\therefore$ Centre $=(7,9)$ and radius $=\frac{6}{\sqrt{2}}=3\sqrt{2}$
$\Rightarrow$ Equation of circle is $|z-(7+9i)|=3\sqrt{2}$