JEE Main · 2014 · 06 Apr 2014 · Offline · Three Dimensional Geometry

The line has direction ratios $(3,1,-5)$ and the plane has normal $(2,-1,1)$. Their dot product is $6-1-5=0$, so the line runs parallel to the plane and its mirror image keeps the very same direction ratios. All that remains is to reflect one point of the line. Reflecting $(1,3,4)$ in $2x-y+z+3=0$ along the normal direction $(2,-1,1)$ lands on $(-3,5,2)$. With the direction unchanged and this point on it, the image is $\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}$.