Problem 35
Question
Is there anything special about the relationship between the lines \(A x+B y=C_{1}\) and \(B x-A y=C_{2}(A \neq 0, B \neq 0) ?\) Give reasons for your answer.
Step-by-Step Solution
Verified Answer
The lines are perpendicular because the product of their slopes is \(-1\).
1Step 1: Identify Slopes of the Lines
The slope of a line given by the equation \(Ax + By = C_1\) can be found by rewriting it in slope-intercept form. Rearranging gives \(y = -\frac{A}{B}x + \frac{C_1}{B}\). Thus, the slope is \(-\frac{A}{B}\). Similarly, rearranging the second line \(Bx - Ay = C_2\), we get \(y = \frac{B}{A}x - \frac{C_2}{A}\), so the slope is \(\frac{B}{A}\).
2Step 2: Analyze the Relationship between Slopes
Check the product of the slopes of the two lines. For line 1, the slope is \(-\frac{A}{B}\), and for line 2, the slope is \(\frac{B}{A}\). The product of the slopes is \(-\frac{A}{B} \times \frac{B}{A} = -1\).
3Step 3: Determine the Special Relationship
Since the product of the slopes of two lines is \(-1\), the lines are perpendicular. Two lines are perpendicular if and only if the slopes are negative reciprocals, which we have shown to be the case.
Key Concepts
Slope of a LineSlope-Intercept FormNegative Reciprocals
Slope of a Line
The slope of a line is a measure of how steep the line is. You can think of it as the "rise over run," which tells you how much the line rises vertically for every unit it moves horizontally. Calculating the slope is essential because it helps us understand the line's direction and how it behaves on a graph.
In mathematical terms, the slope is denoted by the letter "m" and is calculated by the formula \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the \( y \)-coordinate, and \( \Delta x \) is the change in the \( x \)-coordinate between two points on the line.
When you have a linear equation in the form of \( Ax + By = C \), you can rearrange it to slope-intercept form, \( y = mx + b \), to easily determine the slope \( m \).
For example, consider the equation \( Ax + By = C \). Rearranging gives \( y = -\frac{A}{B}x + \frac{C}{B} \), hence the slope \( m \) is \( -\frac{A}{B} \). Understanding the slope is crucial for analyzing the relationship between lines, such as determining if they are parallel or perpendicular.
In mathematical terms, the slope is denoted by the letter "m" and is calculated by the formula \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the \( y \)-coordinate, and \( \Delta x \) is the change in the \( x \)-coordinate between two points on the line.
When you have a linear equation in the form of \( Ax + By = C \), you can rearrange it to slope-intercept form, \( y = mx + b \), to easily determine the slope \( m \).
For example, consider the equation \( Ax + By = C \). Rearranging gives \( y = -\frac{A}{B}x + \frac{C}{B} \), hence the slope \( m \) is \( -\frac{A}{B} \). Understanding the slope is crucial for analyzing the relationship between lines, such as determining if they are parallel or perpendicular.
Slope-Intercept Form
The slope-intercept form of a line, expressed as \( y = mx + b \), is one of the most useful ways to represent a linear equation. This form clearly indicates two vital components of a line on a graph: the slope \( m \) and the y-intercept \( b \).
The slope \( m \), as we discussed earlier, tells us the steepness and direction of the line. The y-intercept \( b \), on the other hand, gives us the point where the line crosses the y-axis. This happens when \( x = 0 \).
By converting equations like \( Ax + By = C \) into slope-intercept form, you simplify the process of plotting them and analyzing their characteristics. For instance, rearranging \( Ax + By = C \) results in \( y = -\frac{A}{B}x + \frac{C}{B} \). Here, the slope is \( -\frac{A}{B} \) and the y-intercept is \( \frac{C}{B} \).
Knowing this form makes it easy to quickly recognize how changes in \( m \) or \( b \) alter the graph of the line, whether it shifts up or down, or rotates about a point.
The slope \( m \), as we discussed earlier, tells us the steepness and direction of the line. The y-intercept \( b \), on the other hand, gives us the point where the line crosses the y-axis. This happens when \( x = 0 \).
By converting equations like \( Ax + By = C \) into slope-intercept form, you simplify the process of plotting them and analyzing their characteristics. For instance, rearranging \( Ax + By = C \) results in \( y = -\frac{A}{B}x + \frac{C}{B} \). Here, the slope is \( -\frac{A}{B} \) and the y-intercept is \( \frac{C}{B} \).
Knowing this form makes it easy to quickly recognize how changes in \( m \) or \( b \) alter the graph of the line, whether it shifts up or down, or rotates about a point.
Negative Reciprocals
Negative reciprocals are a special pair of numbers where the product of the two numbers is -1. In the context of slopes, if you have one slope \( m_1 \), the negative reciprocal will be \( m_2 = -\frac{1}{m_1} \).
This concept is critical when analyzing perpendicular lines. For two lines to be perpendicular, their slopes must be negative reciprocals. This means if one line has a slope of \( -\frac{A}{B} \), the perpendicular line will have a slope of \( \frac{B}{A} \).
This concept is critical when analyzing perpendicular lines. For two lines to be perpendicular, their slopes must be negative reciprocals. This means if one line has a slope of \( -\frac{A}{B} \), the perpendicular line will have a slope of \( \frac{B}{A} \).
- "Negative" implies a change in the sign, so if one slope is positive, the reciprocal should be negative, and vice versa.
- "Reciprocal" refers to flipping the fraction. If it's \( \frac{A}{B} \), its reciprocal would be \( \frac{B}{A} \).
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