Question

Statement-$1$: The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$.
Statement-$2$: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.

• A. Statement-$1$ is true and Statement-$2$ is true; Statement-$2$ is not the correct explanation for Statement-$1$
• B. Statement-$1$ is true, Statement-$2$ is false
• C. Statement-$1$ is false, Statement-$2$ is true
• D. Statement-$1$ is true, Statement-$2$ is true; Statement-$2$ is the correct explanation for Statement-$1$

Answer 1

Answer: (A)

Statement-$1$ is true and Statement-$2$ is true; Statement-$2$ is not the correct explanation for Statement-$1$

The mid-point of $AB=(\frac{3+1}{2},\frac{1+3}{2},\frac{6+4}{2})=(2,2,5)$ lies on the plane.
The direction ratios of $AB$ are given by $(3-1,1-3,6-4)=(2,-2,2)$
The direction ratios of the normal to the plane are given by $(1,-1,1)$.
Hence, $AB$ is the perpendicular bisector of the given plane.
$\Rightarrow$ $A$ is the image of $B$ in the plane.
Statement-2 is correct, the plane does bisect the line joining $A$ and $B$, but that alone is not sufficient to determine whether $A$ and $B$ are mirror images of each other with respect to the plane.
Therefore, statement $2$ is not the correct explanation of statement $1$.
Hence, option 'A' is correct.