Question

Show that the two lines x - 1/2 = y-2/3 = z-3/4 and x - 4/5 = y - 1/2 = z intersect. Find also the point of intersection of these lines.

Answer 1

Given lines are

Any point on line (1) is P (2r + 1, 3r + 2, 4r +3) 

and any point on line (2) is Q (5λ + 4, 2λ + 1, λ) 

Lines (1) and (2) will intersect if P and Q coincide for some value of λ and r. 

∴ 2r + 1 = 5λ + 4 ⇒ 2r – 5λ = 3 ..... (1) 

3r + 2 + 2λ + 1 ⇒ 3r – 2λ = – 1 ..... (2) 

4r + 3 = λ ⇒ 4r – λ = – 3 ..... (3) 

Solving (1) and (2), we get r = – 1, λ = – 1 

Clearly these values of r and λ satisfy eqn. (3) 

Now P ≡ (– 1, – 1, – 1) 

Hence lines (1) and (2) intersect at (– 1, – 1, – 1).