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Q.
The line $\frac{x}{2}=\frac{-y}{3}=\frac{z}{1}$ is vertical. The direction cosines of the line of greatest slope in the plane $3 x-2 y+z=5$ are proportional to
Equation of horizontal plane through origin
$2 x-3 y+z=0$
The direction ratios of the line of intersection of the planes $2 x-3 y+z=0$ and $3 x-2 y+z=5$ are given by $2 a-3 b+c=0$
and $3 a-2 b+c=0$
$\therefore \frac{a}{-1}=\frac{b}{1}=\frac{c}{5}$
Let $l, m, n$ be the $DC's$ of the line of greatest slope, because the line of greatest slope is perpendicular to the line of intersection of the given plane with the horizontal plane.
$-l+m+5 n=0$
and $3 l-2 m+n=0 $
So, $\frac{l}{11}=\frac{m}{16}=\frac{n}{-1}$
$\therefore DC's$ are proportional to $(11,16,-1)$