Statement-1: The point A(1, 0, 7) is the mirror image of the point B(1, 6, 3) in the line: x/1 = (y-

Q: Wait, why do we reject the idea that bisection is enough? If the line bisects the segment, isn't that sufficient for it to be a mirror image?

That is a classic trap! Bisection is a necessary condition, but it is not sufficient. A line can bisect a segment at any angle, but a mirror image requires the line to be the perpendicular bisector. Without the perpendicularity condition, the line could be passing through the midpoint at an oblique angle, which would not create a mirror image.

Q: How do I identify the direction ratios of a line given in the symmetric form $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$?

It is simpler than it looks! The direction ratios of the line are just the denominators $a$, $b$, and $c$. In your problem, the line is $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$, so the direction ratios are $(1, 2, 3)$. Just ensure the coefficients of $x$, $y$, and $z$ are all $1$ before reading the denominators.

Q: Is there a faster way to check if a point is a mirror image without finding the actual image point?

Yes, absolutely! You do not need to find the coordinates of the mirror image point itself. By verifying the two conditions—that the midpoint lies on the line and that the segment $AB$ is perpendicular to the line—you have mathematically proven the mirror image relationship. It is the most efficient path to the solution.

Question

Statement-1: The point $A(1, 0, 7)$ is the mirror image of the point $B(1, 6, 3)$ in the line: $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$. Statement-2: The line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ bisects the line segment joining $A(1, 0, 7)$ and $B(1, 6, 3)$.

Options:

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a 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 a correct explanation for Statement-1.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast 3D coordinate space. You have two points, $A(1, 0, 7)$ and $B(1, 6, 3)$, floating in the void, and a line $L$ defined by:

$$\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$$

This line acts as a mirror. The question is whether $A$ is the mirror image of $B$ across this line. To answer this, we must move beyond simple intuition and embrace the rigorous beauty of 3D geometry.

### The Two Pillars of Reflection

For a point to be a true mirror image of another across a line, the line must act as a **perpendicular bisector**. This gives us two non-negotiable conditions.

First, the midpoint $M$ of the segment $AB$ must lie on the line $L$. If the line does not pass through the midpoint, it cannot be a bisector, let alone a mirror.

Second, the segment $AB$ must be perpendicular to the line $L$. Think of it as the line of sight from the object to its reflection; it must strike the mirror at a perfect $90^{\circ}$ angle. If either condition fails, the mirror image relationship collapses.

### The Midpoint Calculation

Let us calculate the midpoint $M$ of the segment $AB$. Using the midpoint formula:

$$M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)$$

Plugging in our coordinates, we get:

$$M = \left( \frac{1+1}{2}, \frac{0+6}{2}, \frac{7+3}{2} \right) = (1, 3, 5)$$

Now, we test if this point $M(1, 3, 5)$ lies on the line $L$. Substituting these into the line equation:

$$\frac{1}{1} = \frac{3-1}{2} = \frac{5-2}{3} \Rightarrow 1 = 1 = 1$$

The midpoint lies perfectly on the line. Thus, the condition that the line bisects the segment is satisfied.

### The Perpendicularity Test

We must now ensure the line is perpendicular to the segment $AB$. First, we find the direction ratios of the segment $AB$ by calculating the vector $\vec{AB}$:

$$\vec{AB} = (1-1, 6-0, 3-7) = (0, 6, -4)$$

The direction ratios of our line $L$ are $(1, 2, 3)$. For these to be perpendicular, their dot product must be zero.

Calculating the dot product:

$$(0)(1) + (6)(2) + (-4)(3) = 0 + 12 - 12 = 0$$

The dot product is zero. Therefore, the segment $AB$ is indeed perpendicular to the line $L$.

### The Final Verdict

We have satisfied both conditions: the line bisects the segment, and it is perpendicular to it. Therefore, **$A$ is the mirror image of $B$**.

This is the elegance of JEE problems—they test not just your ability to calculate, but your ability to understand the complete logical structure of a geometric proof. Keep practicing, keep visualizing, and let the math guide your intuition.

Select Answer:

Visualized Solution

Visualizing Points $A$ and $B$

Conditions for Mirror Image

Calculating Midpoint $M$

Coordinates of $M$

Checking Statement-2 (Bisection)

Bisection Confirmed

Direction Ratios of $A B$

Checking Perpendicularity

Dot Product Result

Final Conclusion

The Sigma Insight: Equation of a Line in Space

Analyzing the Setup

The Two Pillars of Reflection

The Midpoint Calculation

The Perpendicularity Test

The Final Verdict

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