Key Concepts and Formulas

• Mirror Image of a Point in a Line: A point $A'$ is the mirror image of a point $A$ with respect to a line $L$ if and only if two conditions are met:
  1. The line segment $AA'$ is perpendicular to the line $L$.
  2. The midpoint of the line segment $AA'$ lies on the line $L$. (This implies $L$ is the perpendicular bisector of $AA'$).
• Direction Ratios (DRs):
  • For a line passing through points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the DRs are $(x_2-x_1, y_2-y_1, z_2-z_1)$.
  • For a line in symmetric form $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$, the DRs are $(a, b, c)$.
• Perpendicularity Condition: Two lines with DRs $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if their dot product is zero: $a_1a_2 + b_1b_2 + c_1c_2 = 0$.
• Midpoint Formula: The midpoint of a line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$.

Step-by-Step Solution

We are given two points $A(1,0,7)$ and $B(1,6,3)$, and a line $L: \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$. We need to evaluate the truthfulness of two statements and their relationship.

Part 1: Evaluating Statement - 1

Statement - 1 asserts that point $A(1,0,7)$ is the mirror image of point $B(1,6,3)$ in the line $L$. To verify this, we must check both conditions for a mirror image: perpendicularity and the midpoint lying on the line.

Step 1.1: Determine the Direction Ratios (DRs) of the line segment $AB$.

• Why this step? To check the perpendicularity condition, we need the direction ratios of the line segment connecting the two points.
• Given points are $A(1,0,7)$ and $B(1,6,3)$.
• $DRs_{AB} = (x_B - x_A, y_B - y_A, z_B - z_A) = (1-1, 6-0, 3-7) = (0, 6, -4)$

Step 1.2: Determine the Direction Ratios (DRs) of the given line $L$.

• Why this step? We need the direction ratios of the mirror line to check for perpendicularity.
• The line $L$ is given by $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$.
• Comparing this to the standard symmetric form, the direction ratios of line $L$ are: $DRs_L = (1, 2, 3)$

Step 1.3: Check if line segment $AB$ is perpendicular to line $L$.

• Why this step? This is the first crucial condition for $A$ to be the mirror image of $B$ in $L$.
• Using the perpendicularity condition $a_1a_2 + b_1b_2 + c_1c_2 = 0$ with $DRs_{AB} = (0, 6, -4)$ and $DRs_L = (1, 2, 3)$: $(0)(1) + (6)(2) + (-4)(3) = 0 + 12 - 12 = 0$
• Since the sum is $0$, the line segment $AB$ is indeed perpendicular to the line $L$.

Step 1.4: Find the midpoint of the line segment $AB$.

• Why this step? This is the second crucial condition for $A$ to be the mirror image of $B$ in $L$. The midpoint must lie on line $L$.
• Let $M$ be the midpoint of $AB$. Using the midpoint formula for $A(1,0,7)$ and $B(1,6,3)$: $M = \left(\frac{1+1}{2}, \frac{0+6}{2}, \frac{7+3}{2}\right) = \left(\frac{2}{2}, \frac{6}{2}, \frac{10}{2}\right) = (1, 3, 5)$

Step 1.5: Check if the midpoint $M$ lies on the line $L$.

• Why this step? This verifies the second condition for a mirror image.
• Substitute the coordinates of $M(1,3,5)$ into the equation of line $L: \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$.
• For the x-coordinate: $\frac{1}{1} = 1$
• For the y-coordinate: $\frac{3-1}{2} = \frac{2}{2} = 1$
• For the z-coordinate: $\frac{5-2}{3} = \frac{3}{3} = 1$
• Since all ratios are equal ($1=1=1$), the midpoint $M(1,3,5)$ lies on the line $L$.

Conclusion for Statement - 1: Both conditions for $A$ being the mirror image of $B$ in line $L$ are satisfied. Therefore, Statement - 1 is TRUE.

Part 2: Evaluating Statement - 2

Statement - 2 says that the line $L$ bisects the line segment joining $A(1,0,7)$ and $B(1,6,3)$.

• Why this step? We need to determine if the line $L$ passes through the midpoint of $AB$.
• A line bisects a line segment if and only if it passes through the midpoint of that segment.
• From Step 1.4, we found the midpoint of $AB$ to be $M(1,3,5)$.
• From Step 1.5, we verified that this midpoint $M(1,3,5)$ lies on the line $L$.
• Since the midpoint of $AB$ lies on line $L$, line $L$ bisects the line segment $AB$.

Conclusion for Statement - 2: Statement - 2 is TRUE.

Part 3: Relationship between Statement - 1 and Statement - 2

We have determined that both Statement - 1 and Statement - 2 are true. Now, we must assess if Statement - 2 provides a correct explanation for Statement - 1.

• Statement - 1 (A is the mirror image of B in L) requires two conditions to be met:
  1. The line segment $AB$ must be perpendicular to $L$.
  2. The line $L$ must bisect the line segment $AB$ (i.e., the midpoint of $AB$ lies on $L$).
• Statement - 2 only affirms the second condition: $L$ bisects $AB$.

While $L$ bisecting $AB$ is a necessary condition for $A$ to be the mirror image of $B$ in $L$, it is not a sufficient condition on its own. The perpendicularity condition ($AB \perp L$) is equally essential for a mirror image. Since Statement - 2 does not encompass the perpendicularity aspect, it does not fully explain why Statement - 1 is true.

Therefore, Statement - 2 is not a correct explanation for Statement - 1.

Common Mistakes & Tips

• Distinguish "bisects" from "perpendicular bisector": A line simply bisecting a segment only means it passes through the midpoint. A perpendicular bisector implies both bisection and perpendicularity. The mirror image concept requires the line to be a perpendicular bisector.
• Always check both conditions for mirror image: Neglecting either the perpendicularity or the midpoint condition will lead to an incorrect conclusion.
• Accuracy in calculations: Pay close attention to the calculation of direction ratios and midpoint coordinates, as small errors can propagate.

Summary

We systematically verified both statements. Statement - 1 is true because the line segment $AB$ is perpendicular to line $L$, and its midpoint lies on $L$. Statement - 2 is true because the midpoint of $AB$ lies on line $L$, meaning $L$ bisects $AB$. However, Statement - 2 only covers one of the two conditions required for a mirror image (bisection), omitting the crucial perpendicularity condition. Hence, Statement - 2 is not a complete explanation for Statement - 1.

The final answer is $\boxed{A}$