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

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 Equation Type

The given equation \( y - y_1 = m(x - x_1) \) is in point-slope form, which is a common form of a linear equation where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Step 2: Analyze Slope and Variable Point

In the equation, \( m \) (the slope) and \( x_1 \) (the x-coordinate of the point through which the line passes) are fixed constants. The only changing parameter in the problem is \( y_1 \).

Step 3: Explore the Effects of Changing \( y_1 \)

As \( y_1 \) changes, the equation becomes \( y = m(x - x_1) + y_1 \), indicating different lines with the same slope \( m \) will be drawn for each unique \( y_1 \) value.

Step 4: Derive Implications of a Fixed Slope

Since every line created keeps the same slope \( m \), all these lines will be parallel to each other as long as \( m eq 0 \). Parallel lines have identical slopes but different intercepts.

Step 5: Conclude Based on Multiple Lines

With \( y_1 \) varying but \( m \) constant, we generate a set of lines, each with the same slope. Hence, these lines are parallel, matching the description in option (C).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a versatile tool in algebra that expresses a line when a point and the slope are known. Its general form is given by: \[ y - y_1 = m(x - x_1) \] Here, \( m \) represents the slope of the line, and \( (x_1, y_1) \) is a point through which the line passes.

• The slope \( m \) indicates how steep the line is.
• The coordinates \( (x_1, y_1) \) show a specific point on the line.

Using this form, you can quickly draw or write the equation of a line if you have these two pieces of information. Simply plug them into the formula, and you have your line. You can see how this form directly answers the problem of how to adjust a line to pass through different points with the same slope.

Parallel Lines

When two or more lines in a plane never intersect, these lines are called parallel lines. The most critical aspect of parallel lines is that they have the exact same slope, meaning they rise and run at the same rate. For example, if you have a line with the equation \( y = 2x + 3 \), any line parallel to it will also have a slope of 2.
In the point-slope form, modifying the value of \( y_1 \) while keeping \( m \) constant results in a new line that's parallel to the original. Regardless of how \( y_1 \) changes, the slope \( m \) remains the same, ensuring all lines remain parallel. This is why they never intersect, maintaining equidistance along their entire length.

• Same slope \( m \) ensures parallelism.
• A change in intercept doesn’t affect parallel status.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the simplest and most commonly used formats. This form is expressed as:\[ y = mx + b \]In this equation, \( m \) represents the slope, while \( b \) is the y-intercept, the point where the line crosses the y-axis.

• Slope \( m \) indicates how steep the line is, with \( m = 0 \) indicating a horizontal line.
• The y-intercept \( b \) is valuable for easily graphing the line.

To convert from point-slope to slope-intercept form, solve \[ y - y_1 = m(x - x_1) \] for \( y \). This involves distributing \( m \) and rearranging the terms to isolate \( y \). With the equation in this form, you can easily identify the slope and y-intercept, making graphing or comparing lines straightforward.