The line passing through the points $(5,1, a)$ and $(3, b, 1)$ crosses the $y z-$ plane at the point…
The line passing through the points $(5,1, a)$ and $(3, b, 1)$ crosses the $y z-$ plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$. Then
$a=2, b=8$
$a=4, b=6$
$a=6, b=4$
$a=8, b=2$
Solution
Equation of line passing through $(5,1, a)$ and $(3, b, 1)$ is
$
\frac{x-5}{2}=\frac{y-1}{1-b}=\frac{z-a}{a-1}=\lambda
$
If line crosses $y z-$ plane i.e., $x=0$
$
\begin{aligned}
& x=2 \lambda+5=0 \\
& \Rightarrow \lambda=-5 / 2,
\end{aligned}
$
Since, $y=\lambda(1-b)+1=\frac{17}{2}$
$
\begin{aligned}
& -\frac{5}{2}(1-b)+1=\frac{17}{2} \\
& b=4
\end{aligned}
$
Also, $z=\lambda(a-1)+a=-\frac{13}{2}$
$
\begin{aligned}
& -\frac{5}{2}(a-1)+a=-\frac{13}{2} \\
& \Rightarrow a=6
\end{aligned}
$