Question

If the points P, Q, R, S are (4, 7, 8), (– 1, – 2, 1), (2, 3, 4) and (1, 2, 5) respectively, show that PQ and RS intersect. Also find the point of intersection.

Answer 1

Let the lines PQ and RS intersect at point A.

Let A divide PQ in the ratio λ : 1, then 

From (3), – λk – λ + 4k + 4 = λk + 2λ + k + 2 

or 2λk + 3λ – 3k – 2 = 0 ..... (6) 

From (4), –2λk – 2λ + 7k + 7 = 2λk + 3λ + 2k + 3 

or 4λk + 5λ – 5k – 4 = 0 ..... (7) 

Multiplying equation (6) by 2, and subtracting from equation (7), we get 

– λ + k = 0 or , λ = k 

Putting λ = k in equation (6), we get 

2λ² + 3λ – 3λ – 2 = 0 

or, λ = ± 1. 

But λ ≠ –1, as the co-ordinates of P would then be undefined and in this case 

PQ || RS, which is not true. 

∴ λ = 1 = k. 

Clearly λ k = 1 satisfies eqn. (5). 

Hence our assumption is correct