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Question
Show that the lines `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` intersect each other.also find the coordinates of the point of intersection
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Solution
The variable point on the line `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` is `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` = λ
∴ x + 1 = – 10λ, y + 3 = – λ, z – 4 = λ
∴ x = – 10λ –1, y = – λ – 3, z = λ + 4 .......(i)
Also, the variable point on the line
`(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` is
`(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` = µ
∴ x + 10 = –µ, y + 1 = –3µ, z – 1 = 4µ
∴ x = – µ – 10, y = – 3µ – 1, z = 4µ + 1 .......(ii)
Given lines intersect each other if there exist some values of λ and µ for which
–10λ – 1 = – µ – 10, – λ – 3
= – 3µ –1 and λ + 4
= 4µ + 1
∴ 10λ – µ = 9 .......(iiii)
λ – 3µ = – 2 .......(iv)
λ – 4µ = – 3 .......(v)
Subtracting equation (iv) from (v), we get
λ – 4µ = –3
λ – 3µ = –2
– + = +
– µ = –1
∴ µ = 1
Substituting µ = 1 in (iv), we get
λ – 3(1) = – 2
∴ λ = – 2 + 3
∴ λ = 1
Since the values of λ and µ exist, the given lines intersect each other. To find the point of intersection, substituting the value of λ = 1 in equation (i), we get
x = –10 – 1, y = –1 – 3, z = 1 + 4
∴ x = –11, y = – 4, z = 5
∴ Point of intersection of the lines is (x, y, z) i.e., (–11, – 4, 5).
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