$\text{ The line passing through the points }(5,1,a)\text{ and }(3,b,1)\text{ crosses the }yz\text{-plane at the point }\left(0,\frac{17}{2},\frac{-13}{2}\right)\text{then}$

Equation of the line passing through $\left(5,1,a\right) and \left(3,b,1\right) is \frac{x-5}{2}=\frac{y-1}{1-b}=\frac{z-a}{a-1}$

Let $\frac{x-5}{2}=\frac{y-1}{1-b}=\frac{z-a}{a-1}=\lambda \left(say\right)$

$$\begin{gathered} \Rightarrow x=2\lambda +5, y=\lambda \left(1-b\right)+1, z=\lambda \left(a-1\right)+a \\ The line meets yz-plane at \left(0,\frac{17}{2},-\frac{13}{2}\right) \\ \Rightarrow 2\lambda +5=0, \lambda \left(1-b\right)+1=\frac{17}{2} and \lambda \left(a-1\right)+a=-\frac{13}{2} \\ \Rightarrow \lambda =-\frac{5}{2} and b=4, a=6 \end{gathered}$$