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: Approach
Take the ratio of the middle components as the scale factor between the two direction vectors.

Step 2: Scale
The second components are $3$ and $1$, so line 1's direction vector is $3$ times line 2's.

Step 3: Match components
First: $a-3=3\times5=15$, giving $a=18$. Third: $-5=3(-1-b)$, so $-1-b=-\dfrac53$, giving $b=\dfrac23$.

Step 4: Answer
$(b,a)=\left(\dfrac23,18\right)$, option (C).

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