Animated Solution for Mathematics - Three Dimensional Geometry: 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 $B(1,3,4)$.

Imagine you are standing in front of a mirror. You see yourself, perfectly replicated, staring back. In the world of JEE Advanced, we don't just look at mirrors; we calculate them.

We have two points, $A(3,1,6)$ and $B(1,3,4)$, and a plane defined by the equation $x-y+z=5$. Our goal is to determine if $A$ is the mirror image of $B$ with respect to this plane.

To solve this, we must embrace two fundamental geometric truths. First, the plane must act as a perfect divider, bisecting the line segment $AB$. Second, the line segment $AB$ must be perfectly perpendicular to the plane.

The Midpoint Test

Let's begin with the first condition: bisection. If the plane is a mirror, it must pass through the exact center of the line segment connecting $A$ and $B$. We call this center the midpoint, $M$.

Using the midpoint formula, we calculate:

Performing the arithmetic, we find $M=(2,2,5)$. Now, for the plane to bisect $AB$, this point $M$ must lie on the plane.

We test this by substituting $x=2,y=2,z=5$ into the plane equation $x-y+z=5$. The left-hand side becomes $2-2+5$, which equals $5$. Since $5=5$, the midpoint $M$ lies perfectly on the plane.

The Perpendicularity Test

Now, we move to the second, more subtle condition: perpendicularity. A mirror doesn't just cut a line in half; it must be oriented such that the line connecting the object and its image is perpendicular to the mirror's surface.

To check this, we look at the direction of the line $AB$. The direction ratios of the line joining $A(3,1,6)$ and $B(1,3,4)$ are given by the vector $\overrightarrow{AB}=(1-3,3-1,4-6)=(-2,2,-2)$.

We can simplify this by multiplying by $-1$ to get $(2,-2,2)$. Next, we look at the plane $x-y+z=5$. The coefficients of $x,y,z$ give us the direction ratios of the normal vector $\vec{n}$, which is $(1,-1,1)$.

For the line $AB$ to be perpendicular to the plane, it must be parallel to the normal vector $\vec{n}$. This means their direction ratios must be proportional. Let's check the ratios:

Since all ratios are equal to $2$, the line $AB$ is perfectly parallel to the normal vector. This confirms that the line $AB$ is perpendicular to the plane.

We have successfully verified both conditions. The plane bisects the segment $AB$, and the segment $AB$ is perpendicular to the plane.

Therefore, $A$ is indeed the mirror image of $B$. Statement-1 is true.

Furthermore, because the bisection of the segment is a necessary condition for the mirror image, Statement-2 provides a correct explanation for Statement-1. By breaking down this complex 3D problem into these two elegant, logical steps, we have mastered the geometry of reflection.