If P (2, 3, 4), Q (5, 8, 7) and R (-1, -2, 1) are collinear, then R divides PQ in the ratio:

Question

Question

This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper

Answer 1

Given:

P (2, 3, 4), Q (5, 8, 7) and R (-1, -2, 1) are collinear.

Concept:

If x divides x₁ and x₂in the ration m₁:m₂ then

\(\rm x=\frac{m_1x_2+m_2x_1}{m_1+m_2}\)

Calculation:

P (2, 3, 4), Q (5, 8, 7) and R (-1, -2, 1) are collinear.

Let R divides PQ in ratio k : 1

and x = -1, x₁ = 2 and x₂ = 5

Then x coordinates satisfy the condition

\(\rm x=\frac{kx_2+x_1}{k+1}\)

\(\rm -1=\frac{k(5)+1(2)}{k+1}\)

⇒ - k - 1 = 5k + 2

⇒ 6k = - 3

⇒ k = -1/2

Then the ratio k : 1 = -1 : 2

Ratio can not be negative.

That means R divides PQ in a ratio 1:2 externally.

Hence the option (2) is correct.

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