In ${\mathrm{R}}^3$, consider the planes ${\mathrm{P}}_1\mathrm{ }:\mathrm{y}=0$ and ${\mathrm{P}}_2\mathrm{ }:\mathrm{x}+\mathrm{z}=1.$ Let ${\mathrm{P}}_3$, be a plane, different from ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$, which passes through the intersection of ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$. If the distance of the point $\left(0, 1, 0\right)$, from ${\mathrm{P}}_3$, is $1$, and the distance of a point $(\mathrm{\alpha },\mathrm{ }\mathrm{\beta },\mathrm{ }\mathrm{\gamma })$, from ${\mathrm{P}}_3$, is $2$, then which of the following relations is (are) true ?

• A

  $2\mathrm{\alpha }+\mathrm{\beta }+2\mathrm{\gamma }+2=0$
• Correct answer
  B

  $2\mathrm{\alpha }-\mathrm{\beta }+2\mathrm{\gamma }+4=0$
• C

  $2\mathrm{\alpha }+\mathrm{\beta }-2\mathrm{\gamma }-10=0$
• Correct answer
  D

  $2\mathrm{\alpha }-\mathrm{\beta }+2\mathrm{\gamma }-8=0$

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