JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Three Dimensional Geometry

The line is $\frac{x-1}{-1}=\frac{y+4}{2}=\frac{z+2}{3}$, through $(1,-4,-2)$ with direction $(-1,2,3)$. The plane contains it, so the point lies on it and the direction is perpendicular to the normal:

Adding gives $-2m+n=0$, so $n=2m$ and then $l=8m$; the plane is

At $A(-3,-6,1)$ and $B(2,4,-3)$ the left side takes the values $-28$ and $14$. The point dividing $AB$ internally in the ratio $k:1$ lies on the plane when

so the ratio is $2:1$, giving $k=2$.