Finding Equation Of Line Perpendicular To Another Line

Introduction

In the realm of coordinate geometry, one of the fundamental concepts is the equation of a line. Lines can be represented in various forms, such as slope-intercept form, point-slope form, and standard form. Understanding how to find the equation of a line is crucial for solving a wide range of problems in mathematics and related fields. This article delves into the process of finding the equation of a line that passes through a given point and is perpendicular to another given line. We will explore the underlying principles, the steps involved, and illustrate the process with a detailed example. Mastering this concept is essential for anyone seeking to deepen their understanding of linear equations and their applications.

Understanding Perpendicular Lines

Before we dive into the problem, let's clarify what it means for two lines to be perpendicular. 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 a line has a slope of m, then any line perpendicular to it will have a slope of -1/m, provided m is not zero. This is because the product of the slopes of two perpendicular lines is always -1. This inverse relationship is crucial for determining the slope of the line we seek. Understanding this concept is crucial as we navigate through the solution.

In mathematical terms, if we have two lines, line 1 with slope m1 and line 2 with slope m2, they are perpendicular if and only if m1 * m2* = -1. This relationship stems from the trigonometric properties of right angles and the definition of slope as the tangent of the angle a line makes with the positive x-axis. The negative reciprocal relationship ensures that the lines intersect at a 90-degree angle. For instance, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. Similarly, if a line has a slope of -3, a line perpendicular to it will have a slope of 1/3. This principle is fundamental in solving problems involving perpendicular lines and their equations.

This property extends beyond simple lines; it is a cornerstone in various areas of mathematics and physics. For example, in calculus, the concept of perpendicularity is used to find the normal line to a curve at a given point, which is essential in optimization problems and curve analysis. In physics, perpendicular vectors often represent forces or velocities acting at right angles, which simplifies the analysis of motion and equilibrium. Therefore, a solid grasp of the relationship between slopes of perpendicular lines is not just beneficial for solving geometric problems but also for understanding more advanced concepts in science and engineering. It provides a powerful tool for simplifying complex scenarios and finding elegant solutions.

Problem Statement: Finding the Equation

Our specific problem is to find the equation of a line that passes through the point (-1, 5) and is perpendicular to the line given by the equation y = -3x + 4. This problem combines the concepts of point-slope form and the properties of perpendicular lines. The given line is in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. From this equation, we can directly identify the slope of the given line, which is -3.

The challenge now is to determine the slope of the line perpendicular to this one. Using the principle we discussed earlier, the slope of the perpendicular line will be the negative reciprocal of -3. This means we flip the fraction (which is -3/1) and change the sign, resulting in a slope of 1/3. With the slope of the perpendicular line known, we can proceed to use the given point (-1, 5) to find the equation of the line. The point-slope form of a linear equation is particularly useful in this situation because it allows us to directly incorporate the slope and a point the line passes through.

To recap, the problem requires us to find a new line that not only has a specific slope (the negative reciprocal of the given line's slope) but also passes through a particular point. This is a classic problem in linear algebra and forms the basis for many geometric and analytical problems. The ability to solve this type of problem is crucial for understanding more complex concepts such as tangent lines, normal vectors, and optimization in multivariable calculus. It also has practical applications in fields like computer graphics, where determining perpendicular relationships is essential for rendering shapes and scenes correctly. Therefore, mastering this skill provides a solid foundation for further studies in mathematics and its applications.

Step-by-Step Solution

1. Identify the slope of the given line: The given line is y = -3x + 4, which is in slope-intercept form (y = mx + b). The slope, m, of this line is -3.

2. Determine the slope of the perpendicular line: The slope of a line perpendicular to a line with slope m is -1/m. Therefore, the slope of the perpendicular line is -1/(-3) = 1/3.

3. Use 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 are given the point (-1, 5) and we have found the slope of the perpendicular line to be 1/3. Plugging these values into the point-slope form gives us:

   y - 5 = (1/3)(x - (-1))

   y - 5 = (1/3)(x + 1)

4. Simplify the equation: Now, we simplify the equation to get it into slope-intercept form or standard form. Let's first distribute the 1/3:

   y - 5 = (1/3)x + 1/3

   Next, add 5 to both sides:

   y = (1/3)x + 1/3 + 5

   To add 1/3 and 5, we need a common denominator. 5 can be written as 15/3, so:

   y = (1/3)x + 1/3 + 15/3

   y = (1/3)x + 16/3

5. Convert to standard form (optional): To express the equation in standard form (Ax + By = C), we can multiply the entire equation by 3 to eliminate the fractions:

   3y = x + 16

Analyzing the Answer Choices

Now, let's compare our solution with the given answer choices:

A) y = -3x + 2

B) y = -3x + 8

C) 3y = x + 16

D) 3y = -x + 16

Our simplified equation is 3y = x + 16, which matches answer choice C. This confirms that our solution is correct.

Looking at the other options, we can see why they are incorrect. Options A and B have a slope of -3, which is the slope of the original line, not the perpendicular line. Option D has the correct terms but with the wrong signs, indicating a misunderstanding of the algebraic manipulation required.

Conclusion

In conclusion, the equation of the line passing through the point (-1, 5) and perpendicular to the line y = -3x + 4 is 3y = x + 16. This problem demonstrates the importance of understanding the relationship between the slopes of perpendicular lines and the application of the point-slope form of a linear equation. By following the step-by-step process outlined above, we can confidently solve similar problems. This skill is fundamental in coordinate geometry and has wide-ranging applications in mathematics, physics, and engineering. Mastering these concepts will undoubtedly enhance your problem-solving abilities and deepen your understanding of mathematical principles.

Final Answer

The final answer is (C) $3y=x+16$