Question

Find the value of k for which the line through the points (2, 4, 8) and (1, 2, 4) is parallel to the line through the points (3, 6, k) and (1, 2, 1) ?

1. 10
2. 9
3. 8
4. 0

Answer 1

Correct Answer - Option 2 : 9

Concept:

Let us consider two lines AB and CD. The direction ratios of line AB is a1, b1, c1 and the direction ratios of line CD is a2, b2, c2.

Then AB will be parallel to CD, if  \(\rm \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

Calculation:

Given: The line through the points (2, 4, 8) and (1, 2, 4) is parallel to the line through the points (3, 6, k) and (1, 2, 1).

Let us consider AB be the line joining the points (2, 4, 8) and (1, 2, 4) whereas CD be the line passing through the points (3, 6, k) and (1, 2, 1).

Let, the direction ratios of AB be: a1, b1, c1 

⇒ a₁ = (2 – 1) = 1, b₁ = (4 – 2) = 2 and c₁ = (8 – 4) = 4.

Let the direction ratios of CD be: a2, b2, c2 

⇒ a₂ = (3 – 1) = 2, b₂ = (6 – 2) = 4 and c₂ = k – 1.

∵ Line AB is parallel to CD ⇒  \(\rm \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

⇒ \(\rm \frac{1}{2}=\frac{2}{4}=\frac{4}{k-1}\)

⇒ \(\rm \frac{1}{2}=\frac{4}{k-1}\)

⇒ k - 1 = 8 ⇒ K = 9.

Hence, correct option is 2.