\(\begin{equation} \text { If }\left|\frac{z}{z+i}=2\right| \text { represents a circle with centre } P \text { then dis…

Let \(z=x+i y\)

\(
\begin{aligned}
& |z|=2|z+i| \\
& \sqrt{x^2+y^2}=2 \sqrt{x^2+(y+1)^2} \\
& x^2+y^2=4\left(x^2+(y+1)^2\right) \\
& C: 3 x^2+3 y^2+8 y+4=0 \\
& \therefore \quad P\left(0, \frac{-4}{3}\right)
\end{aligned}
\)

Now \(P D: \sqrt{1^2+\left(5+\frac{4}{3}\right)^2}=\sqrt{1+\frac{361}{9}}=\sqrt{\frac{370}{9}}\)