Finding Points On The Y-Axis Perpendicular To A Line A Coordinate Geometry Problem
In the realm of coordinate geometry, one of the fundamental concepts involves understanding the relationships between lines and points on a coordinate plane. This article delves into a specific problem that requires us to identify a point on the y-axis that lies on a line perpendicular to another given line. To solve this, we will use principles such as finding the slope of a line, determining the equation of a line, and understanding the properties of perpendicular lines. Let's dissect the problem step by step to arrive at the solution.
Problem Statement and Initial Setup
The problem presents a scenario where we have a line, AB, and a point, C. Our objective is to find a point on the y-axis that also lies on a line that passes through point C and is perpendicular to line AB. The provided options suggest that we need to determine the y-intercept of this perpendicular line. This requires us to first understand the slope of line AB, calculate the slope of the line perpendicular to AB, and then use point C to find the equation of the perpendicular line. Coordinate geometry problems often involve these steps, combining algebraic techniques with geometric intuition. The initial setup involves visualizing the coordinate plane and understanding the given points and lines. Key concepts such as the slope-intercept form of a line and the relationship between slopes of perpendicular lines are crucial for solving this problem effectively.
Determining the Slope of Line AB
To start, we need to determine the slope of line AB. The slope, often denoted as m, is a measure of the steepness and direction of a line. It is calculated using the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line. In our problem, we have points A and B (although their coordinates aren't explicitly given in the title, we'll assume we have them or can derive them from additional context). Once we have the coordinates of A and B, we can plug them into the slope formula. For instance, if A were (-10, -6) and B were (-2, -6), the slope would be calculated as (-6 - (-6)) / (-2 - (-10)) = 0 / 8 = 0. This indicates a horizontal line. The slope is a crucial piece of information because it helps us understand the orientation of the line and, more importantly, find the slope of a line perpendicular to it. Understanding the slope not only helps in solving this specific problem but is also a fundamental skill in coordinate geometry. This initial step sets the foundation for the rest of the solution, as the slope of AB will directly influence the slope of the perpendicular line we need to find. Therefore, ensuring the accurate calculation of the slope is paramount. Furthermore, recognizing the implications of different slope values (positive, negative, zero, or undefined) provides a deeper insight into the nature of the line itself. For example, a zero slope indicates a horizontal line, while an undefined slope indicates a vertical line.
Finding the Slope of the Perpendicular Line
Once we have the slope of line AB, the next step is to find the slope of the line perpendicular to AB. The relationship between the slopes of two perpendicular lines is such that their product is -1. If m₁ is the slope of line AB and m₂ is the slope of the line perpendicular to AB, then m₁ * m₂ = -1. Therefore, m₂ = -1 / m₁. For instance, if the slope of AB (m₁) is 2, then the slope of the perpendicular line (m₂) would be -1 / 2. This negative reciprocal relationship is a cornerstone of coordinate geometry and is essential for solving problems involving perpendicular lines. If the slope of AB is 0 (a horizontal line), the slope of the perpendicular line is undefined (a vertical line), and vice versa. This understanding is crucial for correctly determining the equation of the perpendicular line. The concept of perpendicularity is not only vital in mathematics but also has practical applications in various fields such as engineering, architecture, and physics. In these disciplines, ensuring that structures or lines are perpendicular is critical for stability and functionality. Therefore, mastering this concept provides both theoretical and practical advantages. The ability to quickly and accurately determine the slope of a perpendicular line is a valuable skill in problem-solving and critical thinking.
Determining the Equation of the Perpendicular Line
With the slope of the perpendicular line in hand, we can now determine the equation of the perpendicular line. We know that this line passes through point C, which is given as (-6, 0). We can use the point-slope form of a line, which is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. Substituting the coordinates of point C and the slope of the perpendicular line into this equation will give us the equation of the line. For example, if the slope of the perpendicular line is -1/2 and point C is (-6, 0), the equation becomes y - 0 = (-1/2)(x - (-6)), which simplifies to y = (-1/2)x - 3. This equation represents the perpendicular line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The equation of a line is a powerful tool in coordinate geometry, allowing us to describe and analyze linear relationships. Understanding how to derive and manipulate these equations is fundamental to solving a wide range of geometric problems. The ability to translate geometric information into algebraic equations and vice versa is a key skill in mathematical reasoning. Furthermore, the equation of a line provides a concise and precise way to represent an infinite set of points that lie on the line, making it an indispensable tool in mathematical modeling and analysis.
Finding the Y-Intercept
Our ultimate goal is to find the y-intercept, which is the point where the line intersects the y-axis. The y-axis is defined by the equation x = 0. To find the y-intercept, we substitute x = 0 into the equation of the perpendicular line and solve for y. Using our previous example, the equation of the perpendicular line was y = (-1/2)x - 3. Substituting x = 0, we get y = (-1/2)(0) - 3, which simplifies to y = -3. Therefore, the y-intercept is the point (0, -3). In the context of the problem, this means that the point on the y-axis that lies on the line perpendicular to AB and passing through point C is (0, -3). Finding the y-intercept is a common task in coordinate geometry and has various applications, such as determining the initial value in a linear relationship or identifying the point of intersection with the vertical axis. The y-intercept provides valuable information about the line's position and behavior on the coordinate plane. It is also a key parameter in the slope-intercept form of a line, making it an essential concept to understand. The process of finding the y-intercept involves a simple substitution and solution, but it provides a critical piece of the puzzle in many geometric problems.
Analyzing the Options and Selecting the Correct Answer
Now that we've found the y-intercept, the final step is to analyze the options and select the correct answer. The problem provides several options for the point on the y-axis that lies on the perpendicular line. By comparing our calculated y-intercept with the given options, we can identify the correct answer. In our example, we found the y-intercept to be (0, -3). If the options provided included (0, -3), we would select that as the correct answer. If not, we would need to re-evaluate our calculations or assumptions to ensure accuracy. The process of comparing the calculated result with the given options is a crucial step in problem-solving. It allows us to verify our solution and ensure that it aligns with the constraints and conditions of the problem. This step also highlights the importance of careful and accurate calculations throughout the solution process. Selecting the correct answer requires not only the ability to solve the problem but also the ability to interpret the results and relate them back to the original question. This final step reinforces the understanding of the concepts and techniques used in the solution.
Conclusion
In conclusion, solving this coordinate geometry problem involves a series of steps that build upon fundamental concepts such as slope, perpendicular lines, and the equation of a line. By systematically determining the slope of line AB, finding the slope of the perpendicular line, establishing the equation of the perpendicular line, and identifying the y-intercept, we can successfully find the point on the y-axis that meets the given conditions. This problem underscores the importance of understanding the relationships between lines and points in a coordinate plane and the algebraic techniques used to analyze them. The ability to solve such problems is essential for a strong foundation in mathematics and its applications in various fields.
