Consider ellipses $E_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the value of $\sum _{k=1}^{20}\frac{1}{r_k^2}$ is

• Correct answer
  A

  $3080$
• B

  $2870$
• C

  $3210$
• D

  $3320$

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