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Question
If the straight lines `(x - 1)/1 - (y - 2)/2 = (z - 3)/"m"^2` and `(x - 3)/5 = (y - 2)/"m"^2 = (z - 1)/2` are coplanar, find the distinct real values of m
Solution
`|(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, 2, 3), (x2, y2, z2) = (3, 2, 1)
(l1, m1, n1) = (1, 2, m2), (l2, m2, n2) = (1, m2, 2)
`|(3 - 1, 2 - 2, 1 - 3),(1, 2, "m"^2),(1, "m"^2, 2)|` = 0
`|(2, 0, -2),(1, 2, "m"^2),(1, "m"^2, 2)|` = 0
2(4 – m4) – 2(m2 – 2) = 0
8 – 2m4 – 2m2 + 4 = 0
12 – 2m4 – 2m2 = 0
(÷ – 2) – 6 + m4 + m2 = 0
m4 + m2 – 6 = 0
(m2 – 2)(m2 + 3) = 0
m2 – 2 = 2, m² = – 3 (not possible)
m2 = 2
m = `+- sqrt(2)`
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