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Question
Show that the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and `(x - 4)/5 = (y - 1)/2` = z intersect.. Also, find their point of intersection.
Solution
We have lines,
`L_1 : (x - 1)/2 = (y - 2)/3 = (z - 3)/4 = lambda`
And `L_2 : (x - 4)/5 = (y - 1)/2 = z = mu`
Any point on the line L1 is (`2lambda + 1, 3lambda + 2, 4lambda + 3)`
Any point on the line L2 is `(5mu + 4, 2mu + 1, mu)`
If line intersect then there exists a value of λ, μ for which
`(2lambda + 1, 3lambda + 2, 4lambda + 3) = (5mu + 4, 2mu + 1, mu)`
⇒ `2lambda + 1 = 5mu + 4, 3 lambda + 2 = 2mu + 1` and `4lambda + 3 = mu`
Solving fisrt two equations we get `lambda = - 1, mu = -1`
These values of `lambda = - 1, mu = - 1` also satisfy the third equation.
Thus lines interest.
Also the point of intersection is `(-1, -1, -1)`.
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