In ${\mathrm{R}}^3$ , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes ${\mathrm{P}}_1\mathrm{ }:\mathrm{x}+2\mathrm{y}-\mathrm{z}+1=0\mathrm{ }$ and ${\mathrm{P}}_2\mathrm{ }:2\mathrm{x}-\mathrm{y}+\mathrm{z}-1=0.$ Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane ${\mathrm{P}}_1$ . Which of the following points lie (s) on M ?

• Correct answer
  A

  $\left(0,\mathrm{ }-\frac{5}{6},\mathrm{ }-\frac{2}{3}\right)$
• Correct answer
  B

  $\left(-\frac{1}{6},\mathrm{ }-\frac{1}{3},\frac{1}{6}\right)$
• C

  $\left(-\frac{5}{6},\mathrm{ }0,\frac{1}{6}\right)\mathrm{ }$
• D

  $\left(-\frac{1}{3},\mathrm{ }0,\frac{2}{3}\right)$

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