JEE-Advanced 2015 (11/40)

In $\mathbb{R}^3$ , consider the planes $P_1:y=0$ and $P_2:x+z=1$ . Let $P_3$ be a plane, different from $P_1$ and $P_2$ , which passes through the intersection of $P_1$ and $P_2$ . If the distance of the point $(0,1,0)$ from $P_3$ is $1$ and the distance of a point $(\alpha,\beta,\gamma)$ from $P_3$ is $2$ , then which of the following relations is(are) true ? \[\begin{array}{} (1) \, 2\alpha+\beta+2\gamma+2=0 \quad \quad \quad \quad & (2) \, 2\alpha-\beta+2\gamma+4=0 \\ (3) \, 2\alpha+\beta-2\gamma-10=0 & (4) \, 2\alpha-\beta+2\gamma-8=0 \end{array}\]
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $(1),(2)$ , then submit $12$ as the correct answer, if your answer is $(2),(3),(4)$ , then submit $234$ as the correct answer.