A $10$ inches long pencil $AB$ with mid point $C$ and a small eraser $P$ are placed on the horizontal top of a table such that $PC=\sqrt{5}$ inches and $\angle PCB={\tan }^{-1}\left(2\right)$. The acute angle through which the pencil must be rotated about $C$ so that the perpendicular distance between eraser and pencil becomes exactly $1$ inch is :

• Correct answer
  A

  ${\tan }^{-1}\left(\frac{3}{4}\right)$
• B

  ${\tan }^{-1}\left(\frac{1}{2}\right)$
• C

  ${\tan }^{-1}\left(\frac{4}{3}\right)$
• D

  ${\tan }^{-1}\left(1\right)$

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