Question

Consider the planes

P₁ : x – y + z = 1, P₂ : x + y – z = –1, P₃ : x – 3y + 3z = 2.

Let L₁, L₂ and L₃ be the lines of intersection of the planes P₂ and P₃, P₃ and P₁ and P₁ and P₂, respectively.

Statement I: At least two of the lines L₁, L₂ and L₃ are non-parallel.

Statement II: The three planes do not have a common point.

Answer 1

Let l, m, n be the direction ratios of the line L₁.

Then l + m – n = 0

and l – 3m + 3n = 0

Thus, 

Thus, the direction ratios of L₁ are (0, 1, 1)

Similarly, the direction ratios of L₂ and L₃ are (0, 1, 1) and (0, 1, 1), respectively.

Therefore, the lines L₁, L₂ and L₃ are parallel.

So, Statement I is false.

Put z = 0 in P₂: x + y – z = – 1, P₃: x – 3y + 3z = 2

we get,

does not lie on P₁ , so no point of L₁ lies on P₁.

Therefore, the three planes do not have a common point.

Thus, the statement II is true.