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## Homework Statement

Find the value of the parameter α for which the pencil of planes through the straight line AB has a common plane with the pencil of planes through the straight line CD, where A(1, 2α, α), B(3, 2, 1), C(−α, 0, α) and D(−1, 3, −3).

## Homework Equations

Let Δ be a line given by two equations:

A

_{1}x+B

_{1}y+C

_{1}z+D

_{1}=0

A

_{2}x+B

_{2}y+C

_{2}z+D

_{2}=0

The collection of all planes containing a given straight line Δ is called the pencil of planes through Δ.

The plane π belongs to the pencil of planes through the line Δ if and only if there exists λ,μ∈ℝ such that the equation of the plane π is:

λ(A

_{1}x+B

_{1}y+C

_{1}z+D

_{1})+μ(A

_{2}x+B

_{2}y+C

_{2}z+D

_{2})=0

## The Attempt at a Solution

I wrote the equations of the lines AB and CD. But I don't know the condition for a plane to be common to two pencil of planes in the same time.