Finding The Equation Of A Perpendicular Line Slope-Intercept Form Example
The equation of line CD is a quadratic equation y=3x^2-3, not a linear equation. Find the equation of a line perpendicular to line CD, assuming it was intended to be linear, in slope-intercept form, that passes through the point (3,1).
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In the realm of coordinate geometry, understanding the relationships between lines is crucial. One fundamental concept is the perpendicularity of lines, which arises when two lines intersect at a right angle (90 degrees). This article delves into the process of finding the equation of a line perpendicular to a given line, specifically in slope-intercept form, while also passing through a specified point. We will use the equation of line CD, y = 3x² - 3, and the point (3, 1) as a practical example to illustrate the concepts and steps involved.
Understanding the Basics
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Before we dive into the problem, let's solidify our understanding of the fundamental concepts that underpin this mathematical exploration. This foundational knowledge will serve as the bedrock upon which we construct our solution, ensuring a clear and coherent approach. Let's explore these key concepts in detail:
Slope-Intercept Form
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The slope-intercept form is a powerful and widely used way to represent linear equations. This form expresses the equation of a line as y = mx + b, where 'm' denotes the slope of the line and 'b' represents the y-intercept, the point where the line intersects the y-axis. Understanding slope-intercept form is crucial because it provides a clear and concise way to visualize the characteristics of a line, such as its steepness and its point of intersection with the y-axis. The slope, m, quantifies the line's inclination, indicating how much the y-value changes for each unit change in the x-value. A positive slope signifies an upward inclination, while a negative slope indicates a downward inclination. The y-intercept, b, anchors the line's position on the coordinate plane, specifying the exact point where the line crosses the vertical axis. This form's simplicity and intuitive nature make it a fundamental tool in linear algebra and coordinate geometry.
Slope of Perpendicular Lines
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The relationship between the slopes of perpendicular lines is a cornerstone concept in coordinate geometry. Two lines are deemed perpendicular if they intersect at a right angle, forming a 90-degree angle at their point of intersection. The slopes of such lines share a unique relationship: they are negative reciprocals of each other. Mathematically, if one line has a slope of m₁, a line perpendicular to it will have a slope of m₂ = -1/m₁. This inverse relationship is pivotal because it provides a direct method for determining the slope of a line that is perpendicular to a given line. For instance, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. This principle is not merely a mathematical curiosity; it has practical applications in various fields, including engineering, architecture, and computer graphics, where the construction and analysis of right angles are essential. In essence, understanding the negative reciprocal relationship between perpendicular lines' slopes is a fundamental building block in the broader understanding of geometric relationships in the coordinate plane.
Point-Slope Form
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The point-slope form offers an alternative yet equally powerful method for expressing the equation of a line, particularly useful when you have a point on the line and its slope. The point-slope form is expressed as y - y₁ = m(x - x₁)*, where (x₁, y₁) represents a known point on the line, and m is the slope of the line. This form is exceptionally handy in situations where you have a specific point that the line must pass through and the rate at which the line is inclined. It bypasses the need to calculate the y-intercept directly, making it a more streamlined approach in certain scenarios. The point-slope form allows for a direct translation of geometric information – a specific point and a slope – into an algebraic representation. It highlights the line's local behavior around the given point while also encapsulating its overall direction and steepness. This form serves as a bridge between the geometric properties of a line and its algebraic representation, enabling a more flexible and intuitive approach to problem-solving in coordinate geometry. Whether you're constructing lines that meet specific criteria or analyzing existing geometric configurations, the point-slope form is an invaluable tool in your mathematical arsenal.
Problem Breakdown and Solution
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Now, let's tackle the problem at hand. We're given the equation of line CD as y = 3x² - 3 and asked to find the equation of a line perpendicular to it that passes through the point (3, 1). The challenge here is that the given equation is a quadratic, not a linear equation. This means that line CD is actually a curve (a parabola), not a straight line. Therefore, we cannot directly apply the concept of perpendicular slopes for straight lines to this curve across its entire domain.
However, the question likely has a typo and intends to provide a linear equation for line CD. Let's assume the equation was meant to be a linear equation. For the sake of this exercise, let's assume the equation of line CD is y = 3x - 3. This will allow us to proceed with the problem using the principles of linear equations and perpendicular slopes.
Given this correction, the slope of line CD is 3. Therefore, the slope of a line perpendicular to CD will be the negative reciprocal of 3, which is -1/3. Now we know the slope of the perpendicular line and a point it passes through, (3, 1). We can use the point-slope form to find the equation:
y - y₁ = m(x - x₁)
Substitute m = -1/3, x₁ = 3, and y₁ = 1:
y - 1 = (-1/3)(x - 3)
Now, let's convert this to slope-intercept form (y = mx + b):
y - 1 = (-1/3)x + 1
y = (-1/3)x + 1 + 1
y = (-1/3)x + 2
Therefore, the equation of the line perpendicular to line CD and passing through the point (3, 1) is y = (-1/3)x + 2.
Analyzing the Options
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Now, let's examine the options provided in the original question to see which one matches our solution:
A. y = 3x + 0 B. y = 3x - 8 C. y = (-1/3)x + 2 D. y = (-1/3)x + 0
Comparing our solution, y = (-1/3)x + 2, with the given options, we can clearly see that option C is the correct answer. Options A and B have a slope of 3, which is the original slope (assuming the corrected equation), not the negative reciprocal. Option D has the correct slope (-1/3) but the wrong y-intercept.
Key Steps Summarized
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To recap, here are the key steps we followed to solve this problem, which can be applied to similar problems in coordinate geometry:
- Identify the slope of the given line: In our corrected scenario, we identified the slope of line CD (y = 3x - 3) as 3.
- Determine the slope of the perpendicular line: We calculated the negative reciprocal of the given slope, which is -1/3.
- Use the point-slope form: We plugged the perpendicular slope and the given point (3, 1) into the point-slope form: y - y₁ = m(x - x₁).
- Convert to slope-intercept form: We rearranged the equation into y = mx + b to match the desired format.
- Verify the solution: We compared our solution with the provided options to identify the correct answer.
Importance of Correcting Errors and Assumptions
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This problem highlights the importance of identifying and correcting errors or making reasonable assumptions when faced with ambiguous questions. The original equation for line CD (y = 3x² - 3) presented a challenge because it was a quadratic equation, not a linear one. Recognizing this discrepancy and making the assumption that a linear equation was intended allowed us to proceed with the problem using the principles of perpendicular lines. This skill of critical analysis and problem adaptation is invaluable in mathematics and various other fields. It underscores the need to understand the underlying concepts thoroughly so that you can navigate through challenges and arrive at logical solutions, even when the initial problem statement is flawed.
Conclusion
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Finding the equation of a line perpendicular to another line is a fundamental skill in coordinate geometry. This article has provided a comprehensive guide, starting from the basic concepts of slope-intercept form and perpendicular slopes, to the step-by-step solution of a specific problem. By understanding the underlying principles and practicing problem-solving techniques, you can confidently tackle similar challenges. Remember the key steps: identify the given slope, calculate the negative reciprocal for the perpendicular slope, use the point-slope form, and convert to slope-intercept form. With these tools and a keen eye for detail, you'll be well-equipped to navigate the world of linear equations and coordinate geometry.