Question

If the direction ratios of two parallel lines are \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) then \(\dfrac{a_1 c_2}{a_2}=\) ?

• A. \(b_1\)
• B. \(b_2\)
• C. \(b_3\)
• D. \(c_1\)

Hint:

Use the single constant \(k\): \(a_1=k a_2,\ b_1=k b_2,\ c_1=k c_2\). Such substitutions make parallel-lines algebra immediate.

Answer 1

The Correct Option is D

Step 1 (Reason: Parallelism implies a common proportionality constant). Let \[ \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k\neq 0. \] 

Step 2 (Reason: rewrite \(a_1\) and \(c_1\) using \(k\)). From above, \(a_1=k\,a_2\) and \(c_1=k\,c_2\). 

Step 3 (Reason: substitute and cancel). \[ \frac{a_1 c_2}{a_2}=\frac{k\,a_2\cdot c_2}{a_2}=k\,c_2=c_1. \] Hence \(\dfrac{a_1 c_2}{a_2}=c_1\).