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Chapter 2: Problem 71

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+\) \(c_{2}=0\), where \(a \neq 0\) and \(b \neq 0\), are perpendicular to each other.

Short Answer

Expert verified
The statement is true. The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+ c_{2}=0\), where a 鈮� 0 and b 鈮� 0, are perpendicular to each other because their slopes, given by \(m_1 = -\frac{a}{b}\) and \(m_2 = \frac{b}{a}\), are negative reciprocals of each other \((m_1 \times m_2 = -1)\).

Step by step solution

01

Find the slopes of the lines

To find the slope of a line in the form of \(ax + by + c = 0\), we can rearrange it to the slope-intercept form, \(y = mx + b\), where \(m\) is the slope. So let's rearrange both equations: For the first line: \( a x+b y+c_{1}=0 \) \(by = -ax - c_1 \) Slope of first line, \( m_1 = -\frac{a}{b} \) For the second line: \( b x-a y+ c_{2}=0 \) \(ay = bx - c_2 \) Slope of second line, \( m_2 = \frac{b}{a} \)
02

Determine if the slopes are negative reciprocals

Now, we need to check whether the slopes are negative reciprocals or not. Two slopes are negative reciprocals if their product is equal to \(-1\). \(m_1 \times m_2 = (-\frac{a}{b})(\frac{b}{a})\)
03

Simplify the product of the slopes

Simplify the product of the slopes to see if they are equal to \(-1\). \(m_1 \times m_2 = -\frac{a}{b} \times \frac{b}{a} = -\frac{a \times b}{b \times a} = -1\) Since the product of the slopes is equal to \(-1\), the slopes are indeed negative reciprocals. #Conclusion# The statement is true. The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+ c_{2}=0\), where a 鈮� 0 and b 鈮� 0, are perpendicular to each other because their slopes are negative reciprocals of each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
Understanding the slope of a line is crucial when studying coordinate geometry. The slope is a measure of how steep a line is and is usually represented by the letter 'm'. Mathematically, it is defined as the ratio of the rise (the vertical change) to the run (the horizontal change) between any two points on the line. For a line represented by the equation in the standard form, which is \( ax + by + c = 0 \), we can rearrange it into the slope-intercept form to find the slope.

For example, from the rearranged equation \( y = mx + b \), the coefficient of 'x' is the slope. A positive slope means the line is inclining, while a negative slope indicates it's declining. The greater the absolute value of the slope, the steeper the line. Thus, understanding slopes allows you to visualize the direction and steepness of lines in the coordinate plane.
Negative Reciprocals
The concept of negative reciprocals is fundamental when identifying perpendicular lines. For any non-zero number 'a', its reciprocal is 1/a, and a negative reciprocal is simply -1/a. When dealing with the slopes of lines, two lines are perpendicular if their slopes are negative reciprocals of each other.

That is, if the slope of one line is 'm', the other line must have a slope of \( -\frac{1}{m} \) for them to be perpendicular. Whenever you multiply the slopes of two perpendicular lines, the product will always equal -1. This relationship becomes a quick test for perpendicularity when analyzing the equations of lines in coordinate geometry.
Slope-Intercept Form
The slope-intercept form is one of the easiest ways to interpret and graph linear equations. It is written as \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept, the point where the line crosses the y-axis. This form makes it straightforward to identify both the slope and y-intercept directly without any algebraic manipulation.

The importance of the slope-intercept form in geometry cannot be overstated. It simplifies graphing straight lines and understanding their characteristics. Whenever you are given an equation in standard form, converting it to slope-intercept form will facilitate the identification of the line's slope, and in the case of our exercise, allow us to verify the relationship between slopes of possibly perpendicular lines.
Linear Equations
Linear equations form the basis for much of algebra and coordinate geometry. They describe lines in the coordinate plane and can be presented in various forms, including the standard form \( ax + by + c = 0 \) and the slope-intercept form \( y = mx + b \). Besides these, there is also the point-slope form, which is useful for when you know a point on the line and its slope.

In the context of our exercise, understanding how to manipulate and interpret linear equations is essential. It involves being able to rearrange the standard form to the slope-intercept form and using the resulting slope, combined with knowledge about negative reciprocals, to identify perpendicular lines. Recognizing that the product of their slopes should be -1 provides a practical method to confirm perpendicularity.

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Most popular questions from this chapter

The deaths of children less than 1 yr old per 1000 live births is modeled by the function $$ R(t)=162.8 t^{-3.025} \quad(1 \leq t \leq 3) $$ where \(t\) is measured in 50 -yr intervals, with \(t=1\) corresponding to 1900 . a. Find \(R(1), R(2)\), and \(R(3)\) and use your result to sketch the graph of the function \(R\) over the domain \([1,3]\). b. What was the infant mortality rate in \(1900 ?\) In \(1950 ?\) In \(2000 ?\)

A book designer has decided that the pages of a book should have 1 -in. margins at the top and bottom and \(\frac{1}{2}\) -in. margins on the sides. She further stipulated that each page should have an area of 50 in. \(^{2}\). Find a function in the variable \(x\), giving the area of the printed page. What is the domain of the function?

The ratio of working-age population to the elderly in the United States (including projections after 2000 ) is given by $$ f(t)=\left\\{\begin{array}{ll} 4.1 & \text { if } 0 \leq t<5 \\ -0.03 t+4.25 & \text { if } 5 \leq t<15 \\ -0.075 t+4.925 & \text { if } 15 \leq t \leq 35 \end{array}\right. $$ with \(t=0\) corresponding to the beginning of 1995 . a. Sketch the graph of \(f\). b. What was the ratio at the beginning of 2005 ? What will be the ratio at the beginning of 2020 ? c. Over what years is the ratio constant? d. Over what years is the decline of the ratio greatest?

The immigration to the United States from Europe, as a percentage of the total immigration, is approximately \(P(t)=0.767 t^{3}-0.636 t^{2}-19.17 t+52.7 \quad(0 \leq t \leq 4)\) where \(t\) is measured in decades, with \(t=0\) corresponding to the decade of the \(1950 \mathrm{~s}\). a. Complete the table: b. Use the result of part (a) to sketch the graph of \(P\). c. Use the result of part (b) to estimate the decade when the immigration, as a percentage of the total immigration, was the greatest and the smallest.

Find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=16-x^{2}\)
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