Question

If the direction ratios of two parallel lines are \(x,\,5,\,3\) and \(20,\,10,\,6\) then the value of \(x\) is:

• A. \(10\)
• B. \(5\)
• C. \(3\)
• D. \(40\)

Hint:

Parallel (or collinear) direction ratios are always in the same ratio: \((l_1,m_1,n_1)\parallel(l_2,m_2,n_2)\iff \dfrac{l_1}{l_2}=\dfrac{m_1}{m_2}=\dfrac{n_1}{n_2}\).

Answer 1

The Correct Option is A

Step 1 (Reason: Parallel lines have proportional direction ratios). For lines with d.r.s \((x,5,3)\) and \((20,10,6)\), \[ \frac{x}{20}=\frac{5}{10}=\frac{3}{6}. \] 

Step 2 (Reason: simplify known ratios to identify the common factor). \(\dfrac{5}{10}=\dfrac{3}{6}=\dfrac12\), hence the common ratio is \(\dfrac12\). 

Step 3 (Reason: equate first pair to the same ratio and solve for \(x\)). \[ \frac{x}{20}=\frac12 \ \Rightarrow\ x=20\cdot\frac12=10. \] Therefore \(x=10\).