Question

Let $z_1=6+i$ and $z_2=4-3i$. Let $Z$ be a complex number such that the $arg\left(\frac{z-z_1}{z_2-z}\right)=\frac{\pi }{2}$, then $z$ satisfies the realtion

• A. $|z-(5-i)|=5$
• B. $|z-(5-i)|=\sqrt{5}$
• C. $|z-(5+i)|=5$
• D. $|z-(5+i)|=\sqrt{5}$

Answer 1

The correct option is B $|z-(5-i)|=\sqrt{5}$

Arg $\left(\frac{z-z_1}{z_2-z}\right)=\frac{\pi }{2},z-z_1=(z_2-z)\frac{\pi }{2}(z_1-z)=(z_2-z)(-\frac{\pi }{2})$

Centre of semicircle $\frac{z_1+z_2}{2}=5-i\Rightarrow (5-i)$

$\therefore |z-(5-i)|=r$

Where $\Rightarrow r=\frac{z_1+z_2}{2}=\frac{(6+i)-(4-3i)}{2}=\frac{2+4i}{2}=\frac{\sqrt{2^2+4^2}}{2}=\frac{\sqrt{10}}{2}=\sqrt{5}$