For a positive integer $k$, let $\mathrm{P}_k$ be the parabola given by
$y=\frac{x^2}{k}-5$
and let C be the circle given by
$x^2+y^2=25$
Then

• Correct answer
  A

  there is a $k$ such that $\mathrm{P}_k$ and C intersect each other at three distinct points
• Correct answer
  B

  $\mathrm{P}_k$ and C intersect each other at least once for all $k$
• C

  there are infinitely many values of $k$ such that $\mathrm{P}_k$ and C intersect each other at more than one point
• Correct answer
  D

  there is a $k$ such that $\mathrm{P}_k$ and C intersect each other exactly once

Hence, the correct answer is options: 'A', 'B', 'D'.