Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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:

$\frac{a-3}{5}=\frac{3}{1}=\frac{-5}{-b-1}$

From $\frac{a-3}{5}=3$, we get $a-3=15\Rightarrow a=18$.

From $\frac{-5}{-b-1}=3$, we get $-5=-3b-3\Rightarrow 3b=2\Rightarrow b=\frac{2}{3}$.

The values of $b$ and $a$ respectively are $\frac{2}{3},18$.

Answer: $\frac{2}{3},18$