Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Consider the equation \(y-y_{1}=m\left(x-x_{1}\right)\). In this equation, if \(m\) and \(x_{1}\) are fixed and different lines are drawn for different values of \(y^{1}\), then, (A) the lines will pass through a single point (B) there will be one possible line only (C) there will be a set of parallel lines (D) none of these

Step-by-Step Solution

Answer

(C) there will be a set of parallel lines.

Step 1: Identify the Given Equation

The equation given is the point-slope form of a line: \( y - y_{1} = m(x - x_{1}) \). Here, \(m\) is the slope and \((x_{1}, y_{1})\) is a point on the line.

Step 2: Understand the Fixed Parameters

We are told that \(m\) and \(x_{1}\) are fixed, meaning that the slope \(m\) and the x-coordinate of the point are constant for all lines considered.

Step 3: Consider the Changing Parameter

The parameter that changes is \( y_{1} \), meaning for different values of \( y_{1} \), we get different lines.

Step 4: Analyze the Geometric Implication

When only the \( y_{1} \) value changes, the lines \( y - y_{1} = m(x - x_{1}) \) will have the same slope but different y-intercepts. Lines with the same slope are parallel.

Step 5: Select the Correct Option

Since the lines have the same slope and different y-intercepts due to the change in \( y_{1} \), they form a set of parallel lines. Therefore, option (C) is correct.

Key Concepts

Parallel Lines

In geometry, parallel lines are lines in a plane that never meet. This means no matter how far you extend them, they remain a constant distance apart. In the context of linear equations, two lines are parallel if they have the same slope.
Consider when you use the point-slope form of a line, which is given by the equation:

• \( y - y_{1} = m(x - x_{1}) \).

Here, \( m \) represents the slope, and it stays the same for all lines in our given problem. When you have lines with the same slope but different y-intercepts, they do not intersect and are thus parallel.
This concept is perfectly illustrated in the exercise where changing just \( y_{1} \) results in various lines, yet all of them maintain the same slope \( m \), making them parallel to each other. Understanding this property of parallelism helps us determine that option (C) in the exercise is the correct answer.

Linear Equations

Linear equations are fundamental in mathematics, representing lines on a coordinate plane. They can be written in various forms like slope-intercept form, standard form, and point-slope form. Each form has its own advantages depending on the scenario.
In linear equations, the relationship between the x and y coordinates is directly proportional, and the graph of such an equation is a straight line.

• The general form of linear equations is: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• In point-slope form used in the exercise, the equation includes a specific point through which the line passes, \( (x_1, y_1) \), and the slope \( m \).

When solving problems related to linear equations, recognizing these forms can hugely simplify the task of identifying parallel lines or transforming the equation between different formats, aiding in better visualization of lines on the graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most used forms due to its simplicity and clarity. It is represented as:

• \( y = mx + b \).

In this equation:

• \( m \) is the slope of the line, showing how steep the line is.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

Its straightforward representation makes it easy to identify both the slope and y-intercept directly from the equation. This is particularly useful when drawing the line on a graph, as you can start from \( b \) on the y-axis and follow the rise/run given by \( m \).
This form can be derived from other forms like the point-slope form, by solving for \( y \). For example, transforming \( y - y_{1} = m(x - x_{1}) \) into \( y = mx + (y_1 - mx_1) \), gives us a visual understanding of the line's behavior. This flexibility makes the "slope-intercept form" a preferred choice for analyzing and drawing linear relationships in mathematics.