1. Key Concepts and Formulas

• Mirror Image of a Point in a Plane: For a point $A'$ to be the mirror image of another point $A$ in a plane $P$, two conditions must be simultaneously satisfied:
  1. Midpoint Condition: The midpoint of the line segment $AA'$ must lie on the plane $P$. This implies the plane bisects the segment $AA'$.
  2. Perpendicularity Condition: The line segment $AA'$ must be perpendicular to the plane $P$. This means the direction vector of the line $AA'$ must be parallel to the normal vector of the plane $P$.
• Midpoint Formula: The midpoint $M$ of a line segment joining two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by: $M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)$
• Direction Ratios (d.r.'s) of a Line Segment: The direction ratios of a line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are $(x_2-x_1, y_2-y_1, z_2-z_1)$ (or $(x_1-x_2, y_1-y_2, z_1-z_2)$).
• Normal Vector of a Plane: For a plane given by the equation $ax+by+cz=d$, the direction ratios of its normal vector are $(a, b, c)$.
• Condition for Perpendicularity: A line with direction ratios $(l_1, m_1, n_1)$ is perpendicular to a plane with normal direction ratios $(l_2, m_2, n_2)$ if and only if their direction ratios are proportional, i.e., $\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}$.

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2. Step-by-Step Solution

We are given two points, $A(3, 1, 6)$ and $B(1, 3, 4)$, and a plane $x-y+z=5$. We need to evaluate two statements based on these.

Step 1: Analyze Statement-2 – The Plane Bisects the Line Segment AB

Statement-2 claims that the plane $x-y+z=5$ bisects the line segment joining $A(3, 1, 6)$ and $B(1, 3, 4)$. For a plane to bisect a line segment, the midpoint of that segment must lie on the plane.

Step 1.1: Calculate the Midpoint of AB. Let $A = (x_1, y_1, z_1) = (3, 1, 6)$ and $B = (x_2, y_2, z_2) = (1, 3, 4)$. Using the midpoint formula: $M = \left( \frac{3+1}{2}, \frac{1+3}{2}, \frac{6+4}{2} \right)$ $M = \left( \frac{4}{2}, \frac{4}{2}, \frac{10}{2} \right)$ $M = (2, 2, 5)$ The midpoint of the line segment $AB$ is $M(2, 2, 5)$.

Step 1.2: Check if the Midpoint Lies on the Plane. The equation of the given plane is $x-y+z=5$. Substitute the coordinates of the midpoint $M(2, 2, 5)$ into the plane equation: $2 - 2 + 5 = 5$ $5 = 5$ Since the equation holds true, the midpoint $M(2, 2, 5)$ lies on the plane $x-y+z=5$.

Conclusion for Statement-2: The plane $x-y+z=5$ bisects the line segment joining $A$ and $B$. Thus, Statement-2 is True.

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Step 2: Analyze Statement-1 – A is the Mirror Image of B in the Plane

Statement-1 claims that $A(3, 1, 6)$ is the mirror image of $B(1, 3, 4)$ in the plane $x-y+z=5$. For this to be true, both the midpoint condition (which we've verified in Step 1) and the perpendicularity condition must be met.

Step 2.1: Determine the Direction Ratios of the Line Segment AB. Using points $A(3, 1, 6)$ and $B(1, 3, 4)$, the direction ratios of the line segment $AB$ are: $(x_A-x_B, y_A-y_B, z_A-z_B) = (3-1, 1-3, 6-4) = (2, -2, 2)$

Step 2.2: Determine the Direction Ratios of the Normal to the Plane. The equation of the plane is $x-y+z=5$. Comparing this to $ax+by+cz=d$, we have $a=1, b=-1, c=1$. So, the direction ratios of the normal vector to the plane are $(1, -1, 1)$.

Step 2.3: Check for Perpendicularity. For the line segment $AB$ to be perpendicular to the plane, its direction ratios must be proportional to the direction ratios of the plane's normal vector. Direction ratios of $AB = (2, -2, 2)$. Direction ratios of the normal to the plane $= (1, -1, 1)$. We observe that $(2, -2, 2) = 2 \times (1, -1, 1)$. Since the direction ratios are proportional, the line segment $AB$ is perpendicular to the plane $x-y+z=5$.

Conclusion for Statement-1: Both conditions for a mirror image (midpoint on the plane and segment perpendicular to the plane) are satisfied. Therefore, Statement-1 is True.

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Step 3: Evaluate the Relationship Between Statement-1 and Statement-2

We have found that both Statement-1 and Statement-2 are true. Now we need to determine if Statement-2 is a correct explanation for Statement-1.

Statement-2 states that the plane bisects the line segment $AB$. This is the "Midpoint Condition," which is one of the two necessary conditions for $A$ to be the mirror image of $B$ (or vice-versa) in the plane. However, it is not the only condition. The "Perpendicularity Condition" (that the segment $AB$ must be perpendicular to the plane) is also essential.

Since Statement-2 only describes one of the two necessary conditions, it is not a complete or sufficient explanation for Statement-1. While it is a part of the explanation, it does not fully justify why $A$ is the mirror image of $B$.

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3. Common Mistakes & Tips

• Incomplete Check for Mirror Image: A common mistake is to only check the midpoint condition or only the perpendicularity condition. Remember, both are necessary for a point to be a mirror image.
• Sign Errors: Be careful with signs when calculating direction ratios (e.g., $x_2-x_1$ vs $x_1-x_2$) and substituting coordinates into the plane equation.
• Normal Vector Identification: Ensure you correctly identify the coefficients $(a, b, c)$ from the plane equation $ax+by+cz=d$ to get the normal vector's direction ratios.

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4. Summary

We systematically verified both statements. Statement-2, which claims the plane bisects the line segment $AB$, was found to be true by calculating the midpoint and checking if it lies on the plane. Statement-1, which claims $A$ is the mirror image of $B$ in the plane, was also found to be true because both the midpoint condition and the perpendicularity condition (checked by comparing direction ratios of the segment and the plane's normal) were satisfied. Finally, we evaluated the relationship between the statements. Although Statement-2 is true and a necessary part of the definition of a mirror image, it is not a complete explanation for Statement-1 because the perpendicularity condition is also required.

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5. Final Answer

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. The final answer is $\boxed{A}$ which corresponds to option (A).