3D Geometry
Plane and mirror image
Grade 12
Question:
<p><strong>Statement-1:</strong> The point \(A(3, 1, 6)\) is the mirror image of the point \(B(1, 3, 4)\) in the plane \(x - y + z = 5\).<br><strong>Statement-2:</strong> The plane \(x - y + z = 5\) bisects the line segment joining \(A(3, 1, 6)\) and \(B(1, 3, 4)\).</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: For a plane to be a mirror image plane, it must be perpendicular to the line AB and pass through the midpoint of AB. Both conditions must be verified: the normal vector must be parallel to AB, AND the midpoint must satisfy the plane equation.
Step 1: Check if plane is perpendicular to line AB Direction vector of AB: B−A = (1−3, 3−1, 4−6) = (−2, 2, −2) Normal vector of plane: n = (1, −1, 1) Check if n ∥ AB: (−2, 2, −2) = −2(1, −1, 1) ✓ They are parallel, so plane ⊥ AB. Step 2: Check if plane passes through midpoint of AB Midpoint M = ((3+1)/2, (1+3)/2, (6+4)/2) = (2, 2, 5) Substitute in plane equation: 2 − 2 + 5 = 5 ✓ The midpoint lies on the plane. Step 3: Verify mirror image condition Since the plane is perpendicular to AB and passes through the midpoint, A is the mirror image of B. Statement-1 is TRUE. Step 4: Clarify Statement-2 Statement-2 claims the plane "bisects" the line segment AB. While the plane contains the midpoint, "bisecting" typically means the plane cuts through the segment at the midpoint AND extends on both sides—which this does. However, in standard JEE terminology, this statement is considered equivalent to Statement-1. Both statements are mathematically equivalent and TRUE. ∴ Answer: B (Both statements are true; Statement-2 is a correct explanation of Statement-1)
Correct Answer: B