Finding Perpendicular Lines Passing Through The Origin In The Xy-Plane

Jul 13, 2025 by ADMIN 71 views

Navigating the $xy$-Plane Finding the Perpendicular Line Through the Origin

In the realm of coordinate geometry, understanding the relationships between lines is paramount. One common problem involves finding the equation of a line that is perpendicular to a given line and passes through a specific point, often the origin. This article delves into such a problem, providing a step-by-step solution and highlighting the underlying principles of linear equations and perpendicularity. We will explore the concepts of slope, intercepts, and the relationship between the slopes of perpendicular lines. Let's embark on this geometrical journey and unravel the intricacies of finding the equation of a line perpendicular to another, passing gracefully through the origin.

Problem Statement

In the $xy$ -plane, line $l$ gracefully passes through the origin, a point of immense significance, and stands perpendicular to the line whose equation is elegantly expressed as $5x - 2y = 8$ . Our quest is to identify which of the following options could potentially represent an equation that embodies the very essence of line $l$ :

A. $5x - 2y = 8$ B. $y = rac{2}{5}x$ C. $2x + 5y = 0$ D. $y = - rac{5}{2}x + 4$

This problem elegantly combines the fundamental concepts of linear equations, slopes, and perpendicularity. To solve it effectively, we must first determine the slope of the given line and then leverage the relationship between the slopes of perpendicular lines. Let's embark on this mathematical journey and unravel the solution step by step.

Solution

To decipher the equation of line $l$ , we must first embark on a journey to determine the slope of the given line, which is defined by the equation $5x - 2y = 8$ . This equation, in its current form, conceals the secrets of its slope, urging us to unveil it through algebraic manipulation. Our mission is to transform this equation into the slope-intercept form, a form that elegantly reveals the slope and $y$ -intercept of the line. The slope-intercept form, expressed as $y = mx + b$ , where $m$ represents the slope and $b$ signifies the $y$ -intercept, is our key to unlocking the slope of the given line. Let's dive into the algebraic transformations that will lead us to this crucial form.

Step 1 Unveiling the Slope of the Given Line

Embarking on our algebraic quest, we begin by isolating the term containing $y$ on one side of the equation. Starting with $5x - 2y = 8$ , we subtract $5x$ from both sides, resulting in $-2y = -5x + 8$ . Now, the $y$ term stands alone, but it is still bound by a coefficient of $-2$ . To free $y$ completely, we divide both sides of the equation by $-2$ . This division unveils the slope-intercept form of the equation, revealing the slope that lies hidden within.

Performing the division, we obtain $y = rac{-5x + 8}{-2}$ . Simplifying this expression, we arrive at $y = rac{5}{2}x - 4$ . This is the slope-intercept form of the given equation, and it elegantly displays the slope and $y$ -intercept of the line. The coefficient of $x$ , which is $rac{5}{2}$ , is the slope of the given line. This slope is a crucial piece of information, as it holds the key to finding the slope of the line perpendicular to it. Let's delve into the relationship between the slopes of perpendicular lines and discover how we can use this knowledge to our advantage.

Step 2: The Perpendicular Slope

The concept of perpendicularity in coordinate geometry is intimately linked to the slopes of the lines involved. When two lines stand perpendicular to each other, their slopes engage in a unique relationship a dance of reciprocals and negative signs. The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope. This is a fundamental principle that we will use to determine the slope of line $l$ .

The slope of the given line, as we discovered in the previous step, is $rac{5}{2}$ . To find the slope of a line perpendicular to this, we must first find the reciprocal of $rac{5}{2}$ , which is $rac{2}{5}$ . Then, we must negate this reciprocal, resulting in $- rac{2}{5}$ . This value, $- rac{2}{5}$ , is the slope of line $l$ , the line that stands perpendicular to the given line. Now that we have the slope of line $l$ , we are one step closer to identifying its equation. Let's move on to the next step, where we will use the fact that line $l$ passes through the origin to determine its equation completely.

Step 3: The Origin's Role

Line $l$ is not just any line it possesses the special characteristic of passing through the origin. The origin, the point where the $x$ -axis and $y$ -axis intersect, holds the coordinates $(0, 0)$ . This seemingly simple fact provides us with crucial information that will help us pinpoint the equation of line $l$ . When a line passes through the origin, it implies that its $y$ -intercept is zero. In the slope-intercept form of a linear equation, $y = mx + b$ , the $y$ -intercept is represented by $b$ . Therefore, for line $l$ , $b = 0$ .

Knowing that the slope of line $l$ is $- rac{2}{5}$ and its $y$ -intercept is 0, we can construct its equation in slope-intercept form. Substituting these values into the equation $y = mx + b$ , we get $y = - rac{2}{5}x + 0$ , which simplifies to $y = - rac{2}{5}x$ . This is the equation of line $l$ , expressed in slope-intercept form. However, the answer choices provided may not be in this exact form. Therefore, we must compare this equation with the given options and see if any of them are equivalent to it.

Step 4: Spotting the Equation

Now that we have determined the equation of line $l$ to be $y = - rac{2}{5}x$ , our final task is to compare this equation with the answer choices provided and identify the one that matches. Let's examine each option carefully:

A. $5x - 2y = 8$ : This equation represents the original line and is not perpendicular to line $l$ . B. $y = rac{2}{5}x$ : This equation has a slope that is the reciprocal of line $l$ 's slope, but it is not the negative reciprocal. Therefore, this line is not perpendicular to the given line. C. $2x + 5y = 0$ : This equation can be rearranged into slope-intercept form. Subtracting $2x$ from both sides, we get $5y = -2x$ . Dividing both sides by 5, we obtain $y = - rac{2}{5}x$ . This equation matches the equation we derived for line $l$ . D. $y = - rac{5}{2}x + 4$ : This equation has a slope that is the negative reciprocal of the original line's slope, but it does not pass through the origin (its $y$ -intercept is 4).

Upon careful examination, we find that option C, $2x + 5y = 0$ , is equivalent to the equation $y = - rac{2}{5}x$ , which we derived for line $l$ . Therefore, option C is the correct answer.

Final Answer

The equation of line $l$ , which passes through the origin and is perpendicular to the line with equation $5x - 2y = 8$ , is $2x + 5y = 0$ . This corresponds to option C in the given choices.

Therefore, the final answer is (C). $2x + 5y = 0$

Key Concepts Revisited

Slope-Intercept Form

In our journey to unravel the equation of line $l$ , the slope-intercept form of a linear equation, $y = mx + b$ , played a pivotal role. This form elegantly reveals the slope ( $m$ ) and the $y$ -intercept ( $b$ ) of a line, providing us with crucial information for understanding its behavior and position in the coordinate plane. The slope, $m$ , quantifies the steepness of the line, indicating how much the $y$ -value changes for every unit change in the $x$ -value. A positive slope signifies an upward slant, while a negative slope indicates a downward slant. The $y$ -intercept, $b$ , marks the point where the line intersects the $y$ -axis, providing a fixed point of reference for the line's vertical position.

By transforming the given equation, $5x - 2y = 8$ , into slope-intercept form, we were able to extract the slope of the original line, a crucial step in determining the slope of the perpendicular line. The slope-intercept form serves as a powerful tool for analyzing and comparing linear equations, allowing us to quickly grasp their essential characteristics.

Perpendicular Slopes

The concept of perpendicular slopes is a cornerstone of coordinate geometry, governing the relationship between lines that intersect at a right angle. Two lines are deemed perpendicular if and only if the product of their slopes is -1. This relationship can be elegantly expressed by stating that the slope of one line is the negative reciprocal of the slope of the other. In simpler terms, to find the slope of a line perpendicular to a given line, you must first invert the given slope and then change its sign.

This principle was instrumental in solving our problem. Once we determined the slope of the given line to be $rac{5}{2}$ , we applied the concept of perpendicular slopes to find the slope of line $l$ . By taking the negative reciprocal of $rac{5}{2}$ , we arrived at $- rac{2}{5}$ , which is the slope of line $l$ . Understanding the relationship between perpendicular slopes is crucial for solving a wide range of geometry problems, from finding the equations of perpendicular lines to determining the angles between intersecting lines.

The Significance of the Origin

The origin, the point $(0, 0)$ where the $x$ -axis and $y$ -axis intersect, holds a special significance in coordinate geometry. It serves as the point of reference for the entire coordinate plane, and its properties can greatly simplify certain problems. In our case, the fact that line $l$ passes through the origin provided us with a crucial piece of information the $y$ -intercept of line $l$ is 0.

When a line passes through the origin, its equation takes on a simplified form in slope-intercept form, $y = mx + b$ , the $y$ -intercept, $b$ , becomes 0, reducing the equation to $y = mx$ . This simplification made it easier for us to identify the correct equation for line $l$ once we had determined its slope. The origin's unique position and properties make it a valuable tool in solving geometry problems, often providing a shortcut to the solution.

Conclusion

This exploration into finding the equation of a line perpendicular to another and passing through the origin has illuminated the power of combining fundamental concepts in coordinate geometry. We've seen how the slope-intercept form, the relationship between perpendicular slopes, and the properties of the origin can be leveraged to solve geometric problems effectively. Mastering these concepts not only enhances problem-solving skills but also deepens our understanding of the elegant interplay between algebra and geometry. As we continue our mathematical journey, let's remember the lessons learned here and apply them to new challenges, expanding our horizons and solidifying our grasp of the mathematical world.