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

Answer 1

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).