Equation Of A Line Passing Through A Point And Perpendicular To Another Line

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In mathematics, particularly in coordinate geometry, finding the equation of a line that satisfies certain conditions is a fundamental skill. This article delves into a specific scenario: determining the equation of a line that passes through a given point and is perpendicular to another line. We will explore the underlying concepts, the steps involved, and provide a detailed solution to illustrate the process. Understanding this topic is crucial for various applications in mathematics, physics, and engineering.

Understanding the Basics

Before we dive into the problem, let's review some essential concepts:

  • Point-Slope Form: The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. This form is particularly useful when we know a point on the line and its slope.
  • Slope-Intercept Form: The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis). This form is widely used for its simplicity and ease of interpretation.
  • Slope of a Line: The slope of a line measures its steepness and direction. It is defined as the ratio of the change in y to the change in x between any two points on the line. The slope is often denoted by the letter m.
  • Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. If the slope of one line is m, then the slope of a line perpendicular to it is -1/m. This concept is crucial for solving the problem at hand.

Problem Statement

Our goal is to find the equation of a line that passes through the point (8, -2) and is perpendicular to the line whose equation is y = (1/3)x + 2. This problem combines the concepts of point-slope form, slope-intercept form, and the relationship between slopes of perpendicular lines. To solve this, we will first determine the slope of the given line, then find the slope of the line perpendicular to it, and finally use the point-slope form to write the equation of the new line. We will also convert the equation to slope-intercept form to provide a complete solution.

Step-by-Step Solution

Step 1: Identify the Slope of the Given Line

The given line is in slope-intercept form: y = (1/3)x + 2. From this equation, we can directly identify the slope as the coefficient of x. Therefore, the slope of the given line is m1 = 1/3. This is a straightforward application of the slope-intercept form, where the coefficient of x immediately gives us the slope. Recognizing this form is key to quickly extracting the slope, which is necessary for the next steps in solving the problem. The slope indicates the steepness and direction of the line, which is crucial information when determining the properties of lines that are related to it, such as perpendicular lines.

Step 2: Determine the Slope of the Perpendicular Line

Since we want to find a line perpendicular to the given line, we need to find the negative reciprocal of the slope m1. The negative reciprocal of a number is found by flipping the fraction and changing its sign. In this case, the slope of the given line is 1/3. To find the negative reciprocal, we flip the fraction to get 3/1, which is simply 3, and then change the sign to get -3. Therefore, the slope of the line perpendicular to the given line is m2 = -3. This step utilizes the fundamental property that perpendicular lines have slopes that are negative reciprocals of each other, a concept essential for solving problems involving perpendicularity in coordinate geometry.

Step 3: Use the Point-Slope Form

Now that we have the slope of the perpendicular line (m2 = -3) and a point it passes through (8, -2), we can use the point-slope form of a linear equation to write the equation of the line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Substituting the values we have, we get:

y - (-2) = -3(x - 8)

Simplifying this equation, we have:

y + 2 = -3(x - 8)

This is the equation of the line in point-slope form. The point-slope form is particularly useful here because it directly incorporates the given point and the calculated slope, making it a straightforward way to represent the line's equation based on these two pieces of information. It provides a clear representation of the line’s properties in relation to a specific point and its steepness.

Step 4: Convert to Slope-Intercept Form

To express the equation in slope-intercept form (y = mx + b), we need to solve the point-slope equation for y. Starting with the equation we obtained in the previous step:

y + 2 = -3(x - 8)

First, distribute the -3 on the right side:

y + 2 = -3x + 24

Next, subtract 2 from both sides to isolate y:

y = -3x + 24 - 2

y = -3x + 22

This is the equation of the line in slope-intercept form. The slope-intercept form is valuable because it explicitly shows the slope (m = -3) and the y-intercept (b = 22) of the line, providing a clear understanding of how the line behaves on the coordinate plane. Converting to this form often helps in visualizing and analyzing the line's characteristics and its relationship to other lines or points.

Final Answer

The equation of the line passing through (8, -2) and perpendicular to the line y = (1/3)x + 2 is:

  • Point-Slope Form: y + 2 = -3(x - 8)
  • Slope-Intercept Form: y = -3x + 22

This solution demonstrates the process of finding the equation of a line given specific conditions. By understanding the relationships between slopes of perpendicular lines and utilizing the point-slope and slope-intercept forms, we can effectively solve such problems. These skills are essential in various mathematical and scientific contexts, making it a fundamental topic in coordinate geometry. The ability to transition between these forms also highlights the flexibility and interconnectedness of different representations of linear equations.

Conclusion

In conclusion, finding the equation of a line that passes through a specific point and is perpendicular to another line involves a series of logical steps rooted in the principles of coordinate geometry. By understanding the concepts of slope, perpendicularity, and the different forms of linear equations (point-slope and slope-intercept), we can systematically solve these types of problems. The key is to first identify the slope of the given line, then determine the slope of the perpendicular line by finding the negative reciprocal. With the slope and a point on the line, we can use the point-slope form to express the equation, and then convert it to slope-intercept form for clarity and ease of interpretation. This process not only reinforces mathematical skills but also enhances problem-solving abilities applicable in broader contexts. Mastering these concepts is crucial for students and professionals alike, as they form the foundation for more advanced topics in mathematics and related fields. The application of these principles extends beyond theoretical exercises, finding practical use in areas such as physics, engineering, and computer graphics, where understanding spatial relationships and linear equations is paramount.