Question

If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find

1. the ratio in which the point C divides the line segment AB
2. the values of p and q.

Answer 1

Let `veca, vecb, vecc` be the position vectors of A, B and C respectively.

Then `veca = 3hati + 0.hatj + phatk`,

`vecb = -hati + qhatj + 3hatk` and

`vecc = - 3hati + 3hatj + 0.hatk`

(i) As the points A, B, C are collinear, suppose the point C divides line segment AB in the ratio λ:1.

∴ By the section formula,

`vecc = (λ.vecb + 1.veca)/(λ + 1)`

∴ `-3hati + 3hatj + 0.hatk = (λ(-hati + qhatj + 3hatk) + (3hati + 0.hatj + phatk))/(λ + 1)`

∴ `(λ + 1)(- 3hati + 3hatj + 0.hatk) = (- λhati + λqhatj + 3λhatk) + (3hati + 0.hatj + phatk)`

∴ `-3(λ + 1)hati + 3(λ + 1)hatj + 0.hatk = (- λ + 3)hati + λqhatj + (3λ + p)hatk`

By equality of vectors, we have,

– 3 (λ + 1) = – λ + 3    ...(1)

3(λ + 1) = λq      ...(2)

0 = 3λ + p      ...(3)

From equation (1),

– 3λ – 3 = – λ + 3

∴ – 2λ = 6

∴ λ = – 3 

∴ C divides segment AB externally in the ratio 3:1.

(ii) Putting λ = – 3 in equation (2), we get

3(– 3 + 1) = – 3q

∴ – 6 = – 3q

∴ q = 2

Also, putting λ = – 3 in equation (3), we get

0 = – 9 + p

∴ p = 9

Hence p = 9 and q = 2.