Question

Show that the lines `("x" - 4)/1 = ("y" + 3)/-4 = ("z" + 1)/7` and `("x" - 1)/2 = ("y" + 1)/-3 = ("z" + 10)/8` intersect. Find the coordinates of their point of intersection.

Answer 1

Given lines are: 

`("x" - 4)/1 = ("y" + 3)/-4 = ("z" + 1)/7`      ....(i)

`("x" - 1)/2 = ("y" + 1)/-3 = ("z" + 10)/8`    .....(ii)

Let two lines (i) and (ii) intersect at P(α,β ,γ)

∴ P(α,β ,γ) satisfy line (i)

`=> (alpha - 4)/1 = (beta + 3)/-4 = (gamma + 1)/7 = lambda`

⇒ α =λ + 4 , β = -4λ -3  and  γ = 7λ - 1

Again, P(α, β ,γ) satisfy line (i)

`=> (lambda + 4 -1)/2 = (-4lambda - 3 + 1)/-3 = (7lambda - 1 +10)/8`

`=> (lambda + 3)/2 = (-4lambda - 2)/-3 = (7lambda + 9)/8`

`=> -3lambda - 9 = -8lambda - 4 and -32lambda - 16 = -21 lambda - 27`

⇒ λ = 1   and  λ = 1

Since value of λ in both the cases is same.
Thus, both lines (i) and (ii) intersect each other at a point.
And P(λ + 4, -4λ – 3, 7λ – 1) is P(5, -7, 6).
Hence, the coordinates of the point of intersection are (5, -7, 6).