Problem 39 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 \(B(1,6,3)\). (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.

Step-by-Step Solution

Answer

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

Step 1: Verify Statement-1

The line given is \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) which can be parameterized as \( x = 1t, y = 1 + 2t, z = 2 + 3t \). To check if \( A(1,0,7) \) is the mirror image of \( B(1,6,3) \), the midpoint \( M \) of segment \( AB \) should lie on the line.Calculate the midpoint, \( M \), of \( A \) and \( B \):\[M = \left( \frac{1+1}{2}, \frac{0+6}{2}, \frac{7+3}{2} \right) = (1, 3, 5) \]Substitute \( M \) into the parametric equations of the line:For \( x = 1, 1 = 1t \Rightarrow t = 1 \)For \( y = 3, 3 = 1 + 2(1) \Rightarrow t = 1 \)For \( z = 5, 5 = 2 + 3(1) \Rightarrow t = 1 \)\( M \) lies on the line, supporting Statement-1.

Step 2: Verify Statement-2

Statement-2 states the line \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) bisects the segment joining \( A \) and \( B \).We have already computed the midpoint \( M(1, 3, 5) \) of \( AB \).Since \( M \) lies on the line parameterized by\( x = 1t, y = 1 + 2t, z = 2 + 3t \) for \( t = 1 \), the line does indeed bisect the segment \( AB \).Thus, Statement-2 is true.

Step 3: Analyze Correct Explanation

While both statements are true, we must determine if Statement-2 explains Statement-1.Statement-1 is about a reflection or mirror image of a point \( B \) across a line to give point \( A \), which necessitates the midpoint of \( AB \) being on the line.Statement-2 confirms the midpoint falls on the line but does not explain the reflective property.Thus, Statement-2 is not a correct explanation for Statement-1.

Key Concepts

Line Reflection

In 3D geometry, **line reflection** is a transformation that flips a point across a line to its mirror image. To determine if a point, say point \( A \), is the mirror image of another point, \( B \), across a given line, the midpoint of the segment formed by these two points must lie on the line.

• The midpoint ensures that the line bisects the segment perfectly, making any shift symmetric.
• This property holds in 3D space and allows us to verify if such mirroring is valid.

This is crucial because reflection is not just about distance but symmetry; the line must divide the segment equally to verify that point \( A \) is indeed the mirror image of point \( B \). When working with these concepts, ensure that the midpoint satisfies the line's parametric equation. This approach confirms not just the presence of a reflection, but the specific alignment required for exact graphical mirroring.

Coordinate Geometry

**Coordinate Geometry** in three dimensions involves finding points and analyzing lines using three coordinates: \((x, y, z)\) for spatial positioning. This method is a staple of geometry, allowing for accurate placements and transformations in 3D space.

• The coordinates not only identify a point's location but also help track movement and changes, like reflections or translations.
• Parametric equations are a common tool, providing a systematic coordinate representation of lines.

As seen in our earlier example, the line was formulated as \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} \). Here, each coordinate traces a straight path influenced by parameters. These representations make solving geometry problems intuitive, as the math reflects the real-world space clearly and logically. Understanding these fundamentals is invaluable for working with points, lines, or shapes analytically.

Parametric Equations

**Parametric Equations** describe a line in 3D space using one or more parameters, usually denoted by \( t \). This form is particularly useful for analyzing lines' properties and aids in various geometric applications like reflection.
For example, converting the line \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} \) into its parametric format gives us:

• \( x = 1t \)
• \( y = 1 + 2t \)
• \( z = 2 + 3t \)

In this setting, the parameter \( t \) acts as a "locator," showing where any point lies along the line. By substituting values, you can determine positions. In practical use:

• If the midpoint of two points is calculated, it can be checked whether it lies on the line by substituting its coordinates into these equations.
• When all coordinates meet the parameter values consistently, it confirms geometric alignments like midpoint placement or symmetry needed for reflections.

The beauty of parametric equations lies in their ease of use when translating and visualizing geometric ideas in 3D models.