Geometry Level 4

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.

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