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

Let $A=(5,1,a);B=(3,b,1);P=\left(0,\frac{17}{2},\frac{-13}{2}\right)$
The ratio in which the line $\overline{AB}$ divides yz plane $=-x_1:x_2=-5:3$
$\therefore P=\left(\frac{{\displaystyle 5\left(3\right)-3\left(5\right)}}{{\displaystyle 5-3}},\frac{{\displaystyle 5\left(b\right)-3\left(1\right)}}{{\displaystyle 5-3}},\frac{{\displaystyle 5\left(1\right)-3\left(a\right)}}{{\displaystyle 5-3}}\right)$
$\Rightarrow \left(0,\frac{{\displaystyle 17}}{{\displaystyle 2}},\frac{{\displaystyle -13}}{{\displaystyle 2}}\right)=\left(0,\frac{{\displaystyle 5b-3}}{{\displaystyle 2}},\frac{{\displaystyle 5-3a}}{{\displaystyle 2}}\right)$
$\begin{array}{l}\frac{17}{2}=\frac{5b-3}{2}\\ b=4\end{array} \begin{array}{l}\frac{-13}{2}=\frac{5-3a}{2}\\ a=6\end{array}$