Finding The Equation Of A Perpendicular Line In Point-Slope Form

In mathematics, especially in coordinate geometry, finding the equation of a line that satisfies certain conditions is a fundamental problem. This article will delve into a specific scenario: determining the equation of a line in point-slope form that is perpendicular to a given line and passes through a particular point. Understanding this concept is crucial for various applications in mathematics, physics, and engineering. We will break down the process step-by-step, ensuring clarity and comprehension. We will explore the core concepts of slope, perpendicular lines, and the point-slope form equation. By understanding these concepts, we can effectively solve problems involving linear equations and their geometric representations.

The point-slope form is a particularly useful way to represent the equation of a line because it directly incorporates a point on the line and its slope. This form allows us to easily construct the equation of a line given minimal information. Understanding how to manipulate and apply the point-slope form is essential for solving a variety of problems in coordinate geometry and related fields. In this comprehensive exploration, we aim to provide a clear, step-by-step method for determining the equation of a line that is perpendicular to a given line and passes through a specified point, all while expressing the final answer in point-slope form. By mastering this process, readers will gain a valuable tool for tackling linear equation problems with confidence and accuracy. This article will help solidify your understanding of this concept and enable you to apply it effectively in various mathematical contexts.

The point-slope form of a linear equation is a powerful tool for representing a line when you know a point on the line and its slope. The general form is given by:

y - y1 = m(x - x1)

where:

• (x1, y1) is a known point on the line,
• m is the slope of the line.

The slope of a line, often denoted by m, measures the steepness and direction of the line. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Mathematically, if we have two points (x1, y1) and (x2, y2), the slope m is given by:

m = (y2 - y1) / (x2 - x1)

A line with a positive slope rises from left to right, while a line with a negative slope falls from left to right. A horizontal line has a slope of 0, and a vertical line has an undefined slope.

The point-slope form is particularly useful because it directly incorporates a point on the line and its slope into the equation. This makes it easy to write the equation of a line when you have this information. For instance, if you know a line passes through the point (2, 3) and has a slope of 2, you can directly plug these values into the point-slope form:

y - 3 = 2(x - 2)

This equation represents the line that passes through the point (2, 3) and has a slope of 2. The point-slope form provides a straightforward way to represent linear equations, making it a valuable tool for various mathematical and practical applications. Understanding and using the point-slope form effectively can greatly simplify the process of working with linear equations and solving related problems.

Two lines are considered perpendicular if they intersect at a right angle (90 degrees). There is a specific relationship between the slopes of perpendicular lines that is crucial for solving problems like the one presented.

If a line has a slope m1, and another line is perpendicular to it with a slope m2, then the product of their slopes is -1. Mathematically, this relationship is expressed as:

m1 * m2 = -1

This implies that the slope of the perpendicular line is the negative reciprocal of the original line's slope. In other words:

m2 = -1 / m1

For example, if a line has a slope of 2, the slope of a line perpendicular to it would be -1/2. Similarly, if a line has a slope of -3/4, the slope of a perpendicular line would be 4/3. This relationship holds true for all pairs of perpendicular lines, except for horizontal and vertical lines. A horizontal line has a slope of 0, and a vertical line has an undefined slope, so the negative reciprocal relationship doesn't directly apply in this case. However, a horizontal line is always perpendicular to a vertical line.

Understanding this negative reciprocal relationship is key to finding the equation of a line perpendicular to a given line. When you know the slope of the original line, you can easily determine the slope of the perpendicular line. This, combined with a point on the perpendicular line, allows you to use the point-slope form to write the equation of the line. The concept of perpendicularity is fundamental in geometry and has many applications in real-world scenarios, such as construction, navigation, and computer graphics. Mastering the relationship between the slopes of perpendicular lines is essential for solving a wide range of mathematical problems involving linear equations and geometric figures.

Now, let's apply the concepts discussed to solve the problem. We need to find the equation, in point-slope form, of the line that is perpendicular to a given line and passes through the point (2, 5). We are given the following options:

A. y + 5 = x + 2 B. y - 2 = x - 5 C. y - 5 = -(x - 2) D. y + 2 = -(x + 5)

Step 1: Identify the Slope of the Perpendicular Line

First, we need to determine the slope of the line to which our desired line is perpendicular. Since the options are in point-slope form or variations thereof, we can infer the slope of the original line by looking at the coefficient of the (x - x1) term. From the answer choices, we can deduce that the slope of the original line is implicitly 1 (since the coefficient of x is 1 in options A and B, and -1 in options C and D before considering the perpendicularity). Therefore, the slope of the line perpendicular to it will be the negative reciprocal of 1, which is -1.

Step 2: Use the Point-Slope Form

We are given the point (2, 5) through which the perpendicular line passes. Using the point-slope form y - y1 = m(x - x1), where (x1, y1) = (2, 5) and m = -1, we can plug in these values:

y - 5 = -1(x - 2)

Simplifying, we get:

y - 5 = -(x - 2)

Step 3: Match the Equation with the Options

Comparing our result with the given options, we find that it matches option C:

C. y - 5 = -(x - 2)

Therefore, the equation of the line in point-slope form that is perpendicular to the given line and passes through the point (2, 5) is y - 5 = -(x - 2). This step-by-step solution demonstrates how to apply the concepts of point-slope form and perpendicular lines to solve a specific problem. By following this method, you can confidently tackle similar problems involving linear equations and geometric relationships.

Let's delve deeper into why option C, y - 5 = -(x - 2), is the correct answer. This equation is in the point-slope form, which, as we discussed earlier, is given by y - y1 = m(x - x1). In this form, (x1, y1) represents a point on the line, and m is the slope of the line.

In the equation y - 5 = -(x - 2), we can identify the following:

• The point (x1, y1) is (2, 5), as this is the point given in the problem through which the line must pass.
• The slope m is -1, which is the coefficient of the (x - 2) term. This indicates that the line has a negative slope, meaning it falls from left to right.

The negative slope of -1 is crucial because it ensures that the line is perpendicular to the original line. As we established earlier, the slopes of perpendicular lines are negative reciprocals of each other. Since the implicit slope of the original line (deduced from the other options) is 1, the slope of the perpendicular line must be -1. This satisfies the condition for perpendicularity.

Moreover, the equation correctly incorporates the point (2, 5). When x = 2, the term (x - 2) becomes zero, and the equation simplifies to y - 5 = 0, which means y = 5. This confirms that the point (2, 5) lies on the line represented by this equation. Therefore, option C accurately represents a line that passes through the point (2, 5) and is perpendicular to a line with a slope of 1. The combination of the correct slope and the correct point ensures that the equation satisfies all the conditions of the problem, making it the correct solution. Understanding how each component of the point-slope form contributes to the overall equation of the line is essential for both solving problems and gaining a deeper understanding of linear equations.

To further solidify our understanding, let's examine why the other options are incorrect:

• Option A: y + 5 = x + 2

  • This equation can be rewritten in point-slope form as y - (-5) = 1(x - (-2)). This indicates that the line passes through the point (-2, -5) and has a slope of 1. While the slope might seem correct for a line parallel to the original line (which has an implicit slope of 1), it does not satisfy the perpendicularity condition. Additionally, the point (-2, -5) is not the given point (2, 5). Therefore, this option is incorrect because it neither represents a perpendicular line nor passes through the correct point.

• Option B: y - 2 = x - 5

  • This equation implies that the line passes through the point (5, 2) and has a slope of 1. Similar to option A, the slope of 1 does not satisfy the perpendicularity condition. Furthermore, the point (5, 2) is not the given point (2, 5). Thus, this option is incorrect for the same reasons as option A: it does not represent a perpendicular line and does not pass through the specified point.

• Option D: y + 2 = -(x + 5)

  • This equation can be rewritten as y - (-2) = -1(x - (-5)). This indicates that the line passes through the point (-5, -2) and has a slope of -1. While the slope of -1 is correct for a perpendicular line, the point (-5, -2) is not the given point (2, 5). Therefore, this option is incorrect because, although it has the correct slope for a perpendicular line, it does not pass through the specified point.

In summary, options A and B have the incorrect slope and pass through the wrong point. Option D has the correct slope but passes through the wrong point. Only option C, y - 5 = -(x - 2), satisfies both conditions: it has the correct slope for a perpendicular line (-1) and passes through the given point (2, 5). By understanding why each incorrect option fails to meet the required conditions, we reinforce our understanding of the concepts involved and improve our ability to solve similar problems accurately.

In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point involves understanding the relationship between the slopes of perpendicular lines and the point-slope form of a linear equation. The key steps include identifying the slope of the perpendicular line (which is the negative reciprocal of the original line's slope) and then using the point-slope form y - y1 = m(x - x1) to construct the equation.

In the given problem, we successfully determined that the equation of the line perpendicular to a line with an implicit slope of 1 and passing through the point (2, 5) is y - 5 = -(x - 2). This equation satisfies both the condition of perpendicularity (having a slope of -1) and the condition of passing through the point (2, 5). By systematically applying the concepts and steps outlined in this article, you can confidently solve similar problems and gain a deeper understanding of linear equations and their geometric properties.

Understanding the point-slope form and the relationship between slopes of perpendicular lines is not only crucial for solving mathematical problems but also has practical applications in various fields, including engineering, physics, and computer graphics. Mastering these concepts will enhance your problem-solving skills and broaden your understanding of mathematical principles. We hope this comprehensive guide has provided you with the necessary tools and knowledge to tackle such problems with ease and accuracy.