Q78 / 150 · MHT CET 2026 (20 April Shift 2) · Mathematics
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$18, \dfrac{2}{3}$
- B$\dfrac{3}{2}, 18$
- Correct answerC$\dfrac{2}{3}, 18$
- D$-\dfrac{2}{3}, 18$
Solution
Direction ratios of the line joining $(2, 1, 4)$ and $(a - 1, 4, -1)$ are $(a - 1 - 2, 4 - 1, -1 - 4) = (a - 3, 3, -5)$.
Direction ratios of the line joining $(0, 2, b - 1)$ and $(5, 3, -2)$ are $(5 - 0, 3 - 2, -2 - (b - 1)) = (5, 1, -b - 1)$.
Since the lines are parallel, their direction ratios are proportional:
$\dfrac{a - 3}{5} = \dfrac{3}{1} = \dfrac{-5}{-b - 1}$
From $\dfrac{a - 3}{5} = 3$, we get $a - 3 = 15 \Rightarrow a = 18$.
From $\dfrac{-5}{-b - 1} = 3$, we get $-5 = -3b - 3 \Rightarrow 3b = 2 \Rightarrow b = \dfrac{2}{3}$.
The values of $b$ and $a$ respectively are $\dfrac{2}{3}, 18$.
Answer: $\dfrac{2}{3}, 18$
Direction ratios of the line joining $(0, 2, b - 1)$ and $(5, 3, -2)$ are $(5 - 0, 3 - 2, -2 - (b - 1)) = (5, 1, -b - 1)$.
Since the lines are parallel, their direction ratios are proportional:
$\dfrac{a - 3}{5} = \dfrac{3}{1} = \dfrac{-5}{-b - 1}$
From $\dfrac{a - 3}{5} = 3$, we get $a - 3 = 15 \Rightarrow a = 18$.
From $\dfrac{-5}{-b - 1} = 3$, we get $-5 = -3b - 3 \Rightarrow 3b = 2 \Rightarrow b = \dfrac{2}{3}$.
The values of $b$ and $a$ respectively are $\dfrac{2}{3}, 18$.
Answer: $\dfrac{2}{3}, 18$
Hence, the correct answer is option: 'C'.