Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

Statement - 1 : The point \(A(1,0,7)\) is the mirror image of the point \(B(1,6,3)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\). [2011] 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; Statement2 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

(D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Step 1: Understand the Line Equation

The given line is represented by the parametric form \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} = t \). This translates to the parametric equations as \( x = t, \) \( y = 2t + 1, \) \( z = 3t + 2 \). This line has a direction vector \( \overrightarrow{d} = \langle 1, 2, 3 \rangle \) and passes through the point \( (0, 1, 2) \).

Step 2: Find the Perpendicular Foot from Point B

To find the perpendicular foot from point \( B(1,6,3) \) onto the line, the line joining \( B \) and the perpendicular foot \( P(x, y, z) \) will have a direction vector perpendicular to \( \overrightarrow{d} \), implying \( \overrightarrow{BP} \cdot \overrightarrow{d} = 0 \). Substitute \( P = (t, 2t+1, 3t+2) \) into the equation derived by this dot-product condition to find \( t \).

Step 3: Calculate the Foot of Perpendicular

The condition \[ \overrightarrow{BP} \cdot \overrightarrow{d} = 0 \] gives us \( (t - 1) + 2((2t+1) - 6) + 3((3t+2)-3) = 0 \). Solve this to find \( t \), which then gives \( P(x, y, z) = (t, 2t+1, 3t+2) \).

Step 4: Use the Perpendicular Foot to Find Mirror Image

Once we have \( P \), we use \( \overrightarrow{AP} = - \overrightarrow{PB} \) to establish that \( A \) is the reflection of \( B \) about the line. Substitute \( P \) into the evaluations to verify whether \( A(1,0,7) \) and \( B(1,6,3) \) satisfy this condition.

Step 5: Verify Statement-1

After finding \( P \), if the relation \( A = 2P - B \) holds true, then Statement-1 is correct, confirming that \( A \) is the mirror image of \( B \). Do the calculations and verify.

Step 6: Verify Statement-2

To verify Statement-2, calculate the midpoint of the segment \( AB \) and see if the line at \( (0, 1, 2) \) bisects it. The midpoint \( M \) is \( \left(\frac{1+1}{2}, \frac{0+6}{2}, \frac{7+3}{2}\right) \). If this matches any \( P \) found earlier when \( P \) lies on the line, Statement-2 is true.

Step 7: Conclusion: Evaluate Statements

From earlier calculations, determine whether Statement-1 is true based on the reflection theorem, and whether Statement-2 is true based on bisection. Compare results to specified options for the answer.

Key Concepts

Mirror Image

In analytical geometry, the concept of a mirror image involves reflecting a point across a specified line or plane. To find the mirror image of a point across a line, consider both points are equidistant from the line and lie on perpendiculars drawn to the line.

When given a line and a point, determine the foot of the perpendicular from the point to the line. Then, use the relationship between the point, its mirror image, and the foot of the perpendicular to determine the coordinates of the mirror image. In the context of the exercise, point \( A(1,0,7) \) is identified as the mirror image of \( B(1,6,3) \) with respect to a line, meaning point \( A \) and point \( B \) are equidistant from the line, and the line is perfectly perpendicular at the foot of the perpendicular.

Perpendicular Foot

The perpendicular foot from a point to a line in three-dimensional space is the closest point on the line to the original point. This is where a perpendicular from the point would intersect the line. Calculating the perpendicular foot involves finding a point on the line that creates a right angle with a line segment drawn to the original point.

Using direction vectors and dot products, we ensure that the joining vector from the point to the line's foot is perpendicular to the line's direction vector. Solving this mathematical condition, where the dot product equals zero, gives us the parameter (\( t \)) required to determine the exact point on the line. In our exercise, the foot of the perpendicular from \( B(1,6,3) \) is calculated using this method.

Line Intersection

In the context of our exercise, intersecting means understanding at which point the lines or segments cross paths or meet. For a line that bisects a segment joining two points, it means that the mid-point of this segment lies on the line.

To verify if a line bisects a line segment, calculate the midpoint of the segment and check if this point lies on the given line. In our exercise, when considering the line represented by \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} \), it is crucial to see whether this line contains the midpoint of the line segment joining \( A(1,0,7) \) and \( B(1,6,3) \). If it does, the line indeed bisects the segment.

Direction Vectors

Direction vectors are fundamental in understanding lines in three-dimensional space. A direction vector indicates the direction in which a line extends and is instrumental in writing the parametric equations for a line.

Given a line equation such as \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} \), the direction vector is \( \langle 1, 2, 3 \rangle \). This vector determines the direction flow from one point to another along the line.

• Vector component \( 1 \) corresponds to the x-axis.
• Vector component \( 2 \) corresponds to the y-axis.
• Vector component \( 3 \) corresponds to the z-axis.

This direction vector is integral for computing relevant positional relationships, such as establishing perpendicularity or determining reflections, as observed in our exercise.