Find Equation Of A Line Perpendicular

Finding the Equation of a Perpendicular Line: A Comprehensive Guide

Finding the equation of a line perpendicular to another line is a fundamental concept in coordinate geometry. This guide will walk you through the process, explaining the underlying principles and providing numerous examples to solidify your understanding. We'll cover various scenarios, from simple cases to more complex ones involving different forms of the line equation. Understanding this concept is crucial for solving problems in geometry, calculus, and various engineering applications. By the end of this article, you'll be confident in determining the equation of any perpendicular line.

Understanding Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90°). This geometric relationship has a significant algebraic counterpart: the slopes of perpendicular lines are negative reciprocals of each other. This means if one line has a slope m, a line perpendicular to it will have a slope of -1/m. This relationship forms the cornerstone of finding the equation of a perpendicular line.

The Slope-Intercept Form (y = mx + c)

The simplest way to find the equation of a perpendicular line is when the equation of the original line is given in the slope-intercept form, y = mx + c, where m is the slope and c is the y-intercept.

Steps to find the equation of a perpendicular line:

1. Identify the slope of the given line: Determine the value of m in the equation y = mx + c.

2. Calculate the slope of the perpendicular line: Find the negative reciprocal of the slope found in step 1. This is done by changing the sign and inverting the fraction. For example, if the slope of the given line is 2, the slope of the perpendicular line is -1/2. If the slope is -3/4, the slope of the perpendicular line is 4/3.

3. Use the point-slope form: You'll need a point (x₁, y₁) that the perpendicular line passes through. This point could be given in the problem or you might need to find it. Use the point-slope form of a line: y - y₁ = m'(x - x₁), where m' is the slope of the perpendicular line calculated in step 2.

4. Simplify to slope-intercept form (optional): You can rearrange the equation obtained in step 3 into the slope-intercept form (y = m'x + c') for easier visualization.

Example 1:

Find the equation of the line perpendicular to y = 2x + 3 that passes through the point (1, 4).

1. The slope of the given line is m = 2.

2. The slope of the perpendicular line is m' = -1/2.

3. Using the point-slope form with (1, 4): y - 4 = -1/2(x - 1)

4. Simplifying to slope-intercept form: y = -1/2x + 9/2

Example 2:

Find the equation of the line perpendicular to y = -3/4x + 5 and passing through the origin (0, 0).

1. The slope of the given line is m = -3/4.

2. The slope of the perpendicular line is m' = 4/3.

3. Using the point-slope form with (0, 0): y - 0 = 4/3(x - 0)

4. The equation is simply: y = 4/3x

Dealing with Other Forms of Line Equations

Not all line equations are presented in the slope-intercept form. Let's explore how to handle other common forms.

Standard Form (Ax + By = C)

If the equation is in standard form (Ax + By = C), you first need to convert it to the slope-intercept form. To do this, solve for y:

y = (-A/B)x + (C/B)

Now you can follow the steps outlined in the previous section. Remember that the slope is -A/B.

Example 3:

Find the equation of the line perpendicular to 3x + 2y = 6 that passes through (2, 1).

1. Convert to slope-intercept form: 2y = -3x + 6 => y = (-3/2)x + 3. The slope is m = -3/2.

2. The slope of the perpendicular line is m' = 2/3.

3. Using the point-slope form with (2, 1): y - 1 = (2/3)(x - 2)

4. Simplifying: y = (2/3)x - 1/3

Two-Point Form

If you're given two points on the line, you first need to find the slope using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Then, follow the steps for finding the equation of the perpendicular line as before.

Example 4:

Find the equation of the line perpendicular to the line passing through (1, 2) and (3, 6) and passing through (4, 1).

1. Find the slope of the given line: m = (6 - 2) / (3 - 1) = 2

2. The slope of the perpendicular line is m' = -1/2.

3. Using the point-slope form with (4, 1): y - 1 = -1/2(x - 4)

4. Simplifying: y = -1/2x + 3

Horizontal and Vertical Lines

Horizontal and vertical lines present special cases.

• Horizontal lines: A horizontal line has a slope of 0. A line perpendicular to a horizontal line is a vertical line, which has an undefined slope and the equation x = k, where k is a constant.

• Vertical lines: A vertical line has an undefined slope. A line perpendicular to a vertical line is a horizontal line with a slope of 0 and an equation of the form y = k, where k is a constant.

Example 5:

Find the equation of the line perpendicular to y = 5 (a horizontal line) and passing through (2, 3).

The perpendicular line is a vertical line with equation x = 2.

Example 6:

Find the equation of the line perpendicular to x = -1 (a vertical line) and passing through (4, -2).

The perpendicular line is a horizontal line with equation y = -2.

Advanced Scenarios and Applications

The principles discussed above can be extended to more complex scenarios. For example, you might need to find the equation of a line perpendicular to a line segment, or determine the intersection point of two perpendicular lines. These tasks often involve combining the concept of perpendicularity with other geometric and algebraic techniques.

Frequently Asked Questions (FAQ)

Q1: What if the slope of the given line is zero?

A1: If the slope is 0 (horizontal line), the perpendicular line is a vertical line with an undefined slope and the equation x = k, where k is the x-coordinate of any point on the line.

Q2: What if the slope of the given line is undefined?

A2: If the slope is undefined (vertical line), the perpendicular line is a horizontal line with a slope of 0 and the equation y = k, where k is the y-coordinate of any point on the line.

Q3: Can two parallel lines be perpendicular?

A3: No. Parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes. These conditions are mutually exclusive.

Q4: How do I check if my answer is correct?

A4: Substitute the coordinates of the given point into the equation you found. Also, verify that the slope of your line is the negative reciprocal of the original line's slope. You can graph both lines to visually confirm perpendicularity.

Conclusion

Finding the equation of a perpendicular line is a crucial skill in coordinate geometry. By understanding the relationship between the slopes of perpendicular lines and applying the appropriate equation forms, you can confidently solve a wide range of problems. Remember to carefully identify the slope of the given line, calculate the negative reciprocal, and use the point-slope form to find the equation of the perpendicular line. Practice with various examples to solidify your understanding and build your problem-solving skills. With consistent effort, this concept will become second nature to you.