Equation Of A Line Perpendicular To Another Line Through A Point
Introduction
In coordinate geometry, one of the fundamental concepts is the equation of a straight line. Lines can be defined in various forms, such as slope-intercept form, point-slope form, and the general form. Understanding these forms and how to convert between them is crucial for solving problems related to lines. This article will delve into finding the general form of an equation of a line that passes through a given point and is perpendicular to another given line. Specifically, we will explore how to determine the equation of a line that passes through the point A(9, 8) and is perpendicular to the line 7x + 6y = 11. This problem involves several key concepts, including the slopes of perpendicular lines, the point-slope form of a line, and the conversion to the general form of a linear equation.
Understanding the General Form of a Linear Equation
The general form of a linear equation is given by Ax + By + C = 0, where A, B, and C are constants, and A and B are not both zero. This form is particularly useful because it can represent any straight line in the coordinate plane, including vertical and horizontal lines. Unlike the slope-intercept form (y = mx + b), the general form does not explicitly show the slope and y-intercept, but it provides a standardized way to express linear equations. Converting an equation into general form often simplifies certain types of calculations and comparisons. In our problem, we aim to find an equation in this general form that satisfies the given conditions.
The general form offers several advantages in mathematical manipulations. For instance, it is straightforward to compare two lines represented in general form to determine if they are parallel or perpendicular. The coefficients A and B play a critical role in determining the slope of the line, which is given by -A/B. When dealing with multiple lines, having them in general form can simplify the process of finding intersection points or determining the distance between parallel lines. Furthermore, the general form is conducive to more advanced topics such as linear systems and matrix representations of linear equations. Mastering the general form is thus an essential skill in linear algebra and analytic geometry, providing a robust framework for solving a wide range of problems. The process of converting equations from other forms into the general form often involves algebraic manipulations such as rearranging terms, eliminating fractions, and simplifying expressions to arrive at the desired format. Understanding these manipulations is crucial for effectively utilizing the general form in various mathematical contexts.
Key Concepts: Slopes of Perpendicular Lines
One of the most important concepts in coordinate geometry is the relationship between the slopes of perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: if one line has a slope m, the slope of a line perpendicular to it is -1/m. This negative reciprocal relationship is fundamental in solving problems involving perpendicularity. For example, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. This concept is derived from the trigonometric properties of right angles and the definition of slope as the tangent of the angle of inclination.
Understanding the slopes of perpendicular lines is crucial for solving problems in coordinate geometry. The negative reciprocal relationship ensures that the lines intersect at a right angle. Geometrically, this can be visualized by rotating one line 90 degrees about their intersection point; the slope of the rotated line will be the negative reciprocal of the original line’s slope. This relationship is not only important for theoretical understanding but also for practical applications in fields such as engineering, physics, and computer graphics. In engineering, for example, ensuring that structural components are perpendicular is vital for stability and load distribution. In computer graphics, perpendicular lines are used in rendering algorithms to create realistic images and animations. Therefore, a solid grasp of this concept is essential for anyone working with geometric problems or spatial relationships. The negative reciprocal relationship also extends to vectors, where the dot product of two perpendicular vectors is zero, further illustrating the interconnectedness of mathematical concepts.
Finding the Slope of the Given Line
The given line is 7x + 6y = 11. To find its slope, we need to rewrite the equation in slope-intercept form (y = mx + b), where m represents the slope and b is the y-intercept. By rearranging the equation, we can isolate y on one side. Subtracting 7x from both sides gives us 6y = -7x + 11. Then, dividing both sides by 6, we get y = (-7/6)x + 11/6. From this form, we can clearly see that the slope of the given line is -7/6. Identifying the slope is a crucial step, as it allows us to determine the slope of the line perpendicular to it.
To find the slope of the given line, converting it to slope-intercept form is a straightforward method. The slope-intercept form not only reveals the slope but also the y-intercept, which can be useful in other contexts. This process involves basic algebraic manipulation, but it is a fundamental skill in linear algebra. The slope of a line provides a measure of its steepness and direction; a negative slope indicates that the line descends from left to right, while a positive slope indicates that the line ascends. The magnitude of the slope also tells us how steep the line is; a larger magnitude means a steeper line. In the context of our problem, the slope of the given line is a critical piece of information, as it allows us to calculate the slope of the perpendicular line. Understanding how to manipulate linear equations to extract key information like the slope is essential for solving a wide variety of geometric and algebraic problems. The ability to quickly and accurately determine the slope of a line can significantly streamline problem-solving processes.
Determining the Slope of the Perpendicular Line
Since we need to find a line perpendicular to the given line, we use the negative reciprocal relationship. The slope of the given line is -7/6, so the slope of the perpendicular line is the negative reciprocal of -7/6, which is 6/7. This means that our new line will have a slope of 6/7. Knowing the slope and a point that the line passes through allows us to use the point-slope form of a linear equation to define the line.
Determining the slope of the perpendicular line is a key step in solving this problem. The concept of negative reciprocals is central to this process. The negative reciprocal of a number is found by inverting the number and changing its sign. In this case, the negative reciprocal of -7/6 is indeed 6/7. This value represents the slope of any line that is perpendicular to the given line. The slope provides critical information about the line's orientation and steepness, allowing us to construct its equation. This concept is not only useful in theoretical mathematics but also in practical applications such as architecture and engineering, where ensuring perpendicularity is often essential for structural integrity and design. The ability to quickly calculate the negative reciprocal and apply it in geometric contexts is a valuable skill. Understanding this relationship also helps in visualizing the spatial arrangement of lines and their orientations in the coordinate plane.
Using the Point-Slope Form
The point-slope form of a linear equation is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. We have the point A(9, 8) and the slope 6/7. Plugging these values into the point-slope form, we get y - 8 = (6/7)(x - 9). This equation represents the line that passes through the point A(9, 8) and is perpendicular to the given line. However, to fully answer the problem, we need to convert this into the general form.
Using the point-slope form is an efficient method for constructing the equation of a line when a point and the slope are known. The point-slope form is particularly useful because it directly incorporates the given point and slope, making it a straightforward way to define the line. In our case, we have the point A(9, 8) and the slope 6/7. Substituting these values into the point-slope form gives us an equation that accurately represents the line we are looking for. This form highlights the relationship between the coordinates of a point on the line, the slope, and the given point. While the point-slope form is a valid representation of the line, it is often necessary to convert it into other forms, such as the slope-intercept form or the general form, depending on the context of the problem. The point-slope form is a fundamental tool in coordinate geometry, and proficiency in its use is essential for solving a wide range of problems related to lines and their equations.
Converting to General Form
To convert the equation y - 8 = (6/7)(x - 9) to general form, we need to eliminate the fraction and rearrange the terms to the form Ax + By + C = 0. First, multiply both sides of the equation by 7 to eliminate the fraction: 7(y - 8) = 6(x - 9). This simplifies to 7y - 56 = 6x - 54. Now, rearrange the terms to get all terms on one side of the equation: 6x - 7y - 54 + 56 = 0. Simplifying further, we get 6x - 7y + 2 = 0. This is the general form of the equation of the line.
Converting to the general form involves algebraic manipulations to rearrange the equation into the standard Ax + By + C = 0 format. This form is particularly useful for a variety of reasons, including its ability to represent all lines (including vertical lines) and its convenience for certain types of calculations, such as finding the distance from a point to a line. The process of converting to general form often involves eliminating fractions, distributing terms, and combining like terms. In our case, we started with the point-slope form and systematically rearranged the equation to arrive at the general form. The key steps included multiplying by 7 to eliminate the fraction, distributing the constants, and rearranging the terms to get all variables and constants on one side of the equation. The resulting equation, 6x - 7y + 2 = 0, represents the line that passes through the point A(9, 8) and is perpendicular to the given line 7x + 6y = 11. Mastering the conversion to general form is a fundamental skill in linear algebra and coordinate geometry.
Conclusion
In this article, we found the general form of the equation of the line that passes through the point A(9, 8) and is perpendicular to the line 7x + 6y = 11. The general form of the equation is 6x - 7y + 2 = 0. We achieved this by first finding the slope of the given line, then determining the slope of the perpendicular line using the negative reciprocal relationship. Next, we used the point-slope form to create an equation of the line, and finally, we converted the equation to the general form. This process demonstrates the importance of understanding the relationships between different forms of linear equations and the geometric properties of lines.