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Question
If the straight lines `(x - 1)/2 = (y + 1)/lambda = z/2` and `(x + 1)/5 = (y + 1)/2 = z/lambda` are coplanar, find λ and equations of the planes containing these two lines
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Solution
If the two lines are coplanar
`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, "m"_1, "n"_1),("l"_2, "m"_2, "n"_2)|` = 0
(x1, y1, z1) = (1, −1, 0), (x2, y2, z2) = (−1, −1, 0)
(l1, m1, n1) = (2, λ, 2), (l2, m2, n2) = (5, 2, λ)
⇒ `|(-2, 0, 0),(2, lambda, 2),(5, 2, lambda)|` = 0
When λ = 2
(x1, y1, z1) = (1, −1, 0)
(b1, b2, b3) = (2, 2, 2)
(d1, d2, d3) = (5, 2, 2)
`|(x - x_1, y - y_1, z - z_1),("b"_1, "b"_2, "b"_3),("d"_1, "d"_2, "d"_3)|` = 0
⇒ `|(x - 1, y + 1, z - 0),(2, 2, 2),(5, 2, 2)|` = 0
⇒ (x – 1)(0) – (y + 1)(– 6) + z(6) = 0
⇒ 6(y + 1) – 6z = 0
⇒ 6y + 6 – 6z = 0
⇒ y – z + 1 = 0
When λ = 2
(b1, b2, b3) = (2, – 2, 2)
(d1, d2, d3) = (5, 2, – 2)
⇒ `|(x - 1, y + 1, z - 0),(2, -2, 2),(5, 2, -2)|` = 0
⇒ (x – 1)(0) – (y + 1)(– 14) + z(4 + 10) = 0
⇒ 14(y + 1) + 14z = 0
⇒ 14y + 14 + 14z = 0
⇒ y + z + 1 = 0
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