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Question

Find the values of a so that the lines x12=y23=za4;x45=y12=z are skew.

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Solution

Let L1:x12=y23=za4 and; L2:x45=y12=z

The d.r.'s of lines L1 and L2 are respectively 2, 3, 4 and 5, 2, 1

As 2:3:45:2:1, the lines are not parallel.

Now coordinates of any random point on the lines L1 and L2 are respectively

(2λ+1,3λ+2,λ+a) and (5μ+4,2μ+1,μ).

Lines will be skew if, apart from being non-parallel, they do not intersect each other. So, there must not exist a pair of values of λ and μ which satisfy the following three equations

simultaneously : 2λ+1=5μ+4, 3λ+2=2μ+1, 4λ+a=μ

Solving the first two equations, we get : λ=1=μ.

If these values of λ and μ satisfy the third equation then, the lines will be intersecting each other and hence won't be skew lines.

So, for lines L1 & L2 to be skew, we must have 4λ+aμ i.e., 4×(1)+a1 a3.

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