Question

Let $z_1=10+6i$ and $z_2=4+6i$. If $z$ is any complex number such that the argument of $(z-z_1)/(z-z_2)$ is $\pi /4$, then $|z-7-9i|$ is

• A. $\sqrt{2}$
• B. $2\sqrt{2}$
• C. $3\sqrt{2}$
• D. $4\sqrt{2}$

Answer 1

The correct option is C $3\sqrt{2}$



$\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4}$
Locus of $z$ is major arc whose centre is at $z_0$.Applying rotation at $z_0$, we have
$\frac{z_0-(10+6i)}{z_0-(4+6i)}=\frac{|z_0-(10+6i)|}{|z_0-(4+6i)|}{e}^{i\pi /2}$
$\Rightarrow \frac{z_0-(10+6i)}{z_0-(4+6i)}=i$ $\Rightarrow z_0-10-6i=i{z}_0-4i+6$
$\Rightarrow z_0=7+9i$
radius of arc will be $r=3\sqrt{2}$ unit
Hence $|z-7-9i|=|z-z_0|=r=3\sqrt{2}$ unit