Finding Equation Of Perpendicular Line In Point-Slope Form

Jul 14, 2025 by qnaunigon 59 views

Understanding Point-Slope Form and Perpendicular Lines: Finding the Equation

In the realm of coordinate geometry, understanding the relationships between lines is fundamental. Whether it's determining if lines are parallel, perpendicular, or finding the equation of a line that fits specific criteria, these concepts are crucial. This article delves into a common problem in coordinate geometry: finding the equation of a line that is perpendicular to a given line and passes through a specific point. We will explore the point-slope form, a powerful tool for representing linear equations, and how to apply it to solve this type of problem.

Point-Slope Form: A Quick Review

Before diving into the problem, let's quickly recap the point-slope form of a linear equation. This form is particularly useful when you know a point on the line and the slope of the line. The point-slope form is expressed as:

y - y₁ = m(x - x₁)

Where:

y and x are the variables representing the coordinates of any point on the line.
(x₁, y₁) is a known point on the line.
m is the slope of the line.

The slope, often denoted by m, quantifies the steepness and direction of a line. It's defined as the change in y divided by the change in x between any two points on the line. A positive slope indicates an upward slant, while a negative slope indicates a downward slant. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

The point-slope form directly incorporates the slope and a point on the line, making it an efficient way to write the equation of a line when this information is available. It's also a stepping stone to other forms of linear equations, such as slope-intercept form (y = mx + b) and standard form (Ax + By = C).

Perpendicular Lines and Their Slopes

The concept of perpendicular lines is essential for solving the problem at hand. Two lines are perpendicular if they intersect at a right angle (90 degrees). A key property of perpendicular lines is the relationship between their slopes. If two lines are perpendicular, the product of their slopes is -1. Mathematically, if line 1 has a slope of m₁ and line 2 has a slope of m₂, and the lines are perpendicular, then:

m₁ * m₂ = -1

This relationship provides a crucial link between the slope of a given line and the slope of a line perpendicular to it. If you know the slope of one line, you can easily find the slope of a line perpendicular to it by taking the negative reciprocal. The negative reciprocal of a number is found by flipping the fraction and changing its sign. For example, if the slope of a line is 2/3, the slope of a line perpendicular to it is -3/2.

Understanding the relationship between the slopes of perpendicular lines is vital for problems involving geometric constructions, finding shortest distances, and various other applications in mathematics and physics. It allows us to translate geometric concepts into algebraic equations and vice versa.

Problem Statement: Finding the Perpendicular Line

Now, let's tackle the specific problem: What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point $(2,5)$ ? To solve this, we need additional information about the "given line." Let's assume the "given line" has the equation y = x + 3. This assumption allows us to demonstrate the process of finding the equation of a perpendicular line. The key steps involved are:

Determine the slope of the given line: Identify the slope of the line to which the new line must be perpendicular.
Calculate the slope of the perpendicular line: Find the negative reciprocal of the given line's slope.
Apply the point-slope form: Use the slope of the perpendicular line and the given point to write the equation in point-slope form.

By following these steps, we can systematically determine the equation of the line that meets the specified conditions. This process highlights the power of the point-slope form in handling problems involving lines and their properties.

Step 1: Determine the Slope of the Given Line

The "given line" equation is y = x + 3. This equation is in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. By comparing the equation y = x + 3 with the slope-intercept form, we can directly identify the slope. In this case, the coefficient of x is 1, which means the slope of the given line is 1. Understanding how to extract the slope from an equation in slope-intercept form is crucial for various line-related problems. It allows us to quickly assess the steepness and direction of the line. This step is the foundation for finding the slope of the perpendicular line, as we need the original slope to calculate the negative reciprocal.

Step 2: Calculate the Slope of the Perpendicular Line

As discussed earlier, the slopes of perpendicular lines are negative reciprocals of each other. Since the slope of the given line is 1, we need to find the negative reciprocal of 1. The reciprocal of 1 is 1/1, which is still 1. The negative of 1 is -1. Therefore, the slope of the line perpendicular to the given line is -1. This step demonstrates the application of the perpendicular lines' slope relationship, a fundamental concept in coordinate geometry. It highlights how a simple transformation (taking the negative reciprocal) allows us to find the slope of a line that intersects the given line at a right angle. This is a critical step in determining the equation of the perpendicular line.

Step 3: Apply the Point-Slope Form

Now that we have the slope of the perpendicular line (-1) and a point it passes through (2, 5), we can use the point-slope form to write the equation of the line. The point-slope form is y - y₁ = m(x - x₁). We substitute the values m = -1, x₁ = 2, and y₁ = 5 into the equation:

y - 5 = -1(x - 2)

This is the equation of the line in point-slope form. This step demonstrates the power of the point-slope form in directly incorporating the slope and a point on the line to create the equation. The resulting equation represents a line that is perpendicular to the given line and passes through the specified point, fulfilling the requirements of the problem.

Solution and Verification

Therefore, the equation of the line, in point-slope form, that is perpendicular to the line y = x + 3 and passes through the point $(2,5)$ is:

y - 5 = -(x - 2)

This corresponds to option C in the original problem statement. To verify the solution, we can consider the properties of perpendicular lines and the given point. The equation represents a line with a slope of -1, which is the negative reciprocal of the given line's slope (1), confirming perpendicularity. Additionally, substituting the point (2, 5) into the equation should satisfy the equation:

5 - 5 = -(2 - 2)
0 = 0

The equation holds true, confirming that the line passes through the point (2, 5). This verification process reinforces the correctness of the solution and demonstrates a thorough approach to problem-solving in coordinate geometry.

Conclusion

This article has demonstrated how to find the equation of a line perpendicular to a given line and passing through a specific point, using the point-slope form. We revisited the point-slope form, the relationship between slopes of perpendicular lines, and the step-by-step process of solving this type of problem. Understanding these concepts is crucial for various applications in mathematics, physics, and engineering. The ability to manipulate linear equations and understand their geometric interpretations is a valuable skill in problem-solving and analytical thinking. By mastering these concepts, you can confidently tackle a wide range of coordinate geometry problems and gain a deeper appreciation for the elegance and power of mathematical tools.