JEE Main · 2011 · 10 Apr 2011 · Offline · Three Dimensional Geometry

For $A$ to be the mirror image of $B$ in the line, the segment $AB$ must be perpendicular to the line and its midpoint must lie on the line. The midpoint of $A(1,0,7)$ and $B(1,6,3)$ is $(1,3,5)$, and substituting into $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ gives $1=1=1$, so it lies on the line, making the second statement true. The direction of $AB$ is $(0,6,-4)$, and its dot product with the line's direction $(1,2,3)$ is $0+12-12=0$, so $AB$ is perpendicular too, making the first statement true. But merely bisecting the segment does not by itself guarantee a mirror image without the perpendicularity, so the second statement is not the correct explanation of the first.