Question

If Lines \( \frac{x-1}{-3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2} \) and \( \frac{x-1}{3 \lambda}=\frac{y-1}{2}=\frac{z-6}{-5} \) are perependicular, 7 and value g \( \lambda \). Find \( t \) Lines intersect or not?

Answer 1

For lines to be perpendicular, we have

a₁a₂ + b₁b₂ + c₁c₂ = 0 

⇒ -3 x 3λ + 2λ  + 2 + 2 x -5 = 0

⇒ -9λ + 4λ - 10 = 0

⇒ -5λ + 4λ - 10 = 0

⇒ -5λ = 10

⇒ λ = 10/-5 = -2

∴ Given lines are

\(\frac{x-1}{-3} = \frac{y-2}{-4}=\frac{z-3}2=s\)

and \(\frac{x-1}{-6} = \frac{y-2}{2}=\frac{z-2}{-5}=t\)

⇒ x = 1 - 3s, y = 2 - 4s, z = 3 + 2s

∴ (1 - 3s, 2 - 4s, 3 + 2s) will lie on line (1)

 & x = 1 - 6t, y = 1 + 2t, z = 6 - 5t

∴ (1 - 6t, 1 + 2t, 6 - 5t) will lie on line (2)

If both lines intersects, then

1 - 3s = 1 - 6t

⇒ 3s - 6t = 0 ⇒ s = 2t

& 2 - 4s = 1 + 2t ⇒ 4s = 1 - 2t ⇒ 8t = 1 - 2t

⇒ t = 1/10

∴ s = 2/10 = 1/5

and 3 + 2s = 6 - 5t

⇒ 3 + 2/5 = 6 - 5/10

⇒ 17/5 = 11/2 (Not satisfies)

∴ Both lines did not intersect each other.