Problem 107 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A symmetrical form of the line of intersection of the planes \(x=a y+b\) and \(z=c y+d\) is \(\quad\) [Online April 12, 2014] (a) \(\frac{x-b}{a}=\frac{y-1}{1}=\frac{z-d}{c}\) (b) \(\frac{x-b-a}{a}=\frac{y-1}{1}=\frac{z-d-c}{c}\) (c) \(\frac{x-a}{b}=\frac{y-0}{1}=\frac{z-c}{d}\) (d) \(\frac{x-b-a}{b}=\frac{y-1}{0}=\frac{z-d-c}{d}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(\frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c}\).

Step 1: Identify the Parametric Equations

The parametric form of the plane equations can be given as follows:- **Plane 1**: \( x = a y + b \)- **Plane 2**: \( z = c y + d \)These equations suggest that the intersecting line can be represented using \( y \) as the parameter.

Step 2: Set Up Parametric Form for Line of Intersection

To find the line where these planes intersect, express each variable in terms of parameter \( y \):- Using Plane 1: \( x = a y + b \)- Using Plane 2: \( z = c y + d \)- Parameter: \( y = y \)

Step 3: Write Symmetrical Form of the Line

The symmetrical form of a line involves expressing each variable equated to a common parameter (in this case, \( y \)) in terms of fractions that relate to the direction ratios:- From equations in Step 2, the direction ratios for \( x, y, z \) are \( a, 1, c \) respectively.- The symmetrical form of the line is thus: \[\frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c}\]

Step 4: Check Against the Given Options

Compare the derived symmetrical form with the options provided:- Option (a) matches the derived form: \(\frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c}\)Thus, this option (a) is the correct answer.

Key Concepts

Parametric Equations

Parametric equations are essential tools in mathematics for describing lines. Instead of expressing equations in a traditional manner, parametric equations use a parameter, often denoted as \( t \) or another variable like \( y \), to express the coordinates of points on a line. This allows for a complete representation of a line's position at any point.

In the context of planes intersecting, each plane is typically presented in its parametric form to find their line of intersection. For instance, if you have two planes with equations \( x = ay + b \) and \( z = cy + d \), \( y \) can be used as the parameter to describe any point on the line where these planes intersect.

This method gives us a set of equations relating \( x, y, \) and \( z \) to the parameter \( y \), forming a line in three-dimensional space. Understanding parametric equations is a step into more advanced topics in coordinate geometry, providing flexibility in handling lines and curves.

Symmetrical Form

The symmetrical form of a line is another way to represent the line, commonly used in analytical geometry. This form is particularly helpful when you're dealing with lines in relation to planes or other geometric entities.

In a three-dimensional context, the symmetrical form can be given as a set of equal fractions, each describing one of the coordinate components \( (x, y, z) \) in relation to the line's direction ratios. For example, the symmetrical form \( \frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c} \) signifies that \( y \) is the controlling parameter and \( a, 1, \) and \( c \) are the ratios according to the direction of the line.

This form clearly shows how each point along the line aligns with the given direction vectors, serving as a direct tool for visualization and calculation. The symmetrical form is powerful in geometric problems due to its simple fraction-based layout that ties together three dimensions in a unified way.

Direction Ratios

Direction ratios play a key role in determining the orientation of a line in space. These are a set of three numbers that are proportional to one another and indicate the direction along a line. Typically, if you're given a line in a parametric or symmetrical form, the numbers corresponding to changes in the \( x, y, \) and \( z \) directions are the direction ratios.

For example, in the symmetrical form of a line \( \frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c} \), the direction ratios are \( a, 1, \) and \( c \). These values illustrate how much \( x, y, \) and \( z \) change relative to each other as you move along the line. This could mean for every unit increase in \( y \), \( x \) increases by \( a \) units, and \( z \) by \( c \) units.

Direction ratios are foundational in determining angles between lines and planes and are crucial in engineering and physics, where they help in modeling real-world problems with precision.