If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$, then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

• A

  $4 r$
• B

  $r^2$
• C

  $2 r^2$
• Correct answer
  D

  $4 r^2$

Hence, the correct answer is option: 'D'.