Question

Let the locus of any point $P\left(z\right)$ in the argand plane is $arg\left(\frac{z - 5 i}{z + 5 i}\right)=\frac{\pi }{4}.$ If $O$ is the origin, then the value of $\frac{m a x . \left(O P\right) + m i n . \left(O P\right)}{2}$ is

• A. $5\sqrt{2}$
• B. $5+\frac{5}{\sqrt{2}}$
• C. $5+5\sqrt{2}$
• D. $10-\frac{5}{\sqrt{2}}$

Answer 1

Answer: (B)

Since, $r^{2}+r^{2}=10^{2}$
$\Rightarrow r=5\sqrt{2}$
$max.\left(O P\right)=OC+$ radius
$=5+5\sqrt{2}$
and $min.\left(O P\right)=OA=5$
Required value $=\frac{5 + 5 + 5 \sqrt{2}}{2}=5+\frac{5}{\sqrt{2}}$ .