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प्रश्न
If the lines `(x - 1)/(2) = (y + 1)/(3) = (z -1)/(4) and (x- 2)/(1) = (y +m)/(2) = (z - 2)/(1)` intersect each other, find m.
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उत्तर
The lines `(x - 1)/(2) = (y + 1)/(3) = (z -1)/(4) and (x- 2)/(1) = (y +m)/(2) = (z - 2)/(1)` intersect, if
∴ `|(x_2 - x_1,y_2 - y_2,z - z_2),(a_1, b_1, c_1),(a_2, b_2, c_2)|` = 0 ...(1)
Here, (x1, y1, z1) ≡ (1, – 1, 1),
(x2, y2, z2) ≡ (2, – m, 2),
a1 = 2, b1 = 3, c1 = 4,
a2 = 1, b2 = 2, c2 = 1
Substituting these values in (1), we get
`|(2 - 1, -m + 1, 2 - 1),(2, 3, 4),(1, 2, 1)|` = 0
∴ `|(1, 1 - m, 1),(2, 3, 4),(1, 2, 1)|` = 0
∴ 1(3 – 8) – (1 – m)(2 – 4) + 1(4 – 3) = 0
∴ –5 + 2 – 2m + 1 = 0
∴ – 2m = 2
∴ m = – 1.
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