Question

If the line joining points \((2,1,4)\) and \((a-1,4,-1)\) is parallel to the line joining points \((0,2,b-1)\) and \((5,3,-2)\) then the values of \(b\) and \(a\) are respectively

• A. \(18,\frac{2}{3}\)
• B. \(\frac{3}{2},18\)
• C. \(\frac{2}{3},18\)
• D. \(-\frac{2}{3},18\)

Hint:

Parallel lines have proportional direction ratios.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Two lines are parallel when their direction ratios are proportional.

Step 2: Direction ratios
Line 1 through \((2,1,4)\) and \((a-1,4,-1)\): \((a-3,\ 3,\ -5)\).
Line 2 through \((0,2,b-1)\) and \((5,3,-2)\): \((5,\ 1,\ -1-b)\).

Step 3: Proportion
\[ \frac{a-3}{5}=\frac31=\frac{-5}{-1-b} \]

Step 4: Solve for a
\[ a-3=15\Rightarrow a=18 \]

Step 5: Solve for b
\[ \frac{-5}{-1-b}=3\Rightarrow-1-b=-\frac53\Rightarrow b=\frac23 \]
So \(b=\dfrac23\) and \(a=18\), option (C). The order in the options is b first, then a.

Final Answer:
Proportional direction ratios give a = 18 and b = 2/3, option (C). \[ \boxed{b=\frac23,\ a=18} \]