Equation of a Line Perpendicular to a Given Line

Understanding the concept of perpendicular lines is crucial in geometry and algebra. In this article, we will explore the method to find the equation of a line perpendicular to a given line using specific examples.

Introduction to Perpendicular Lines

A line is considered perpendicular to another if their slopes are negative reciprocals of each other. If the slope of one line is (m), then the slope of a line perpendicular to it will be (-frac{1}{m}).

Example 1: Finding the Equation of a Perpendicular Line

Given the equation of a line (2x - 4y - 3 0), we aim to find the equation of the line that is perpendicular to this line and passes through the point ((2, -1)).

Step 1: Determine the Slope of the Given Line

First, let's convert the given line equation to slope-intercept form (y mx b).

2x - 4y - 3  04y  2x - 3y  frac{1}{2}x - frac{3}{4}

The slope (m) of the given line is (frac{1}{2}).

Step 2: Find the Slope of the Perpendicular Line

The slope of the line perpendicular to the given line will be the negative reciprocal of (frac{1}{2}), which is (-2).

Step 3: Use the Point-Slope Form to Find the Equation of the Perpendicular Line

The point-slope form of the equation of a line is (y - y_1 m(x - x_1)), where (m) is the slope and ((x_1, y_1)) is a point on the line.

Plugging in the slope (-2) and the point ((2, -1)), we get:

y - (-1)  -2(x - 2)y   1  -2x   4y  -2x   3

Alternatively, we can write this in standard form as:

2x   y - 3  0

Conclusion

The equation of the line perpendicular to (2x - 4y - 3 0) and passing through the point ((2, -1)) is (y -2x 3) or (2x y - 3 0).

Additional Examples

Let's explore two more examples to solidify our understanding.

Example 2: Perpendicular Line with Known Coefficients

Given the line (2x - 4y - 3 0), we want to find the equation of the line perpendicular to it that passes through the point ((2, -1)).

Step 1: Determine the Slope of the Given Line

The slope of the given line is (frac{1}{2}), as derived in the previous example.

Step 2: Find the Slope of the Perpendicular Line

The slope of the perpendicular line is (-2).

Step 3: Determine the Equation Using the New Line Form

The new line equation in standard form is (4x 2y - A 0). Substituting the point ((2, -1)) into this equation, we get:

4(2)   2(-1) - A  08 - 2 - A  0A  6

Thus, the equation of the perpendicular line is:

4x   2y - 6  0

Example 3: Another Point and Line

Given a line with the equation (2x 4y 3 0), we need to find the equation of the line perpendicular to it passing through the point ((2, -1)).

Step 1: Determine the Slope of the Given Line

Rewriting the line in slope-intercept form:

2x   4y   3  04y  -2x - 3y  -frac{1}{2}x - frac{3}{4}

The slope of the given line is (-frac{1}{2}).

Step 2: Find the Slope of the Perpendicular Line

The slope of the perpendicular line is (2).

Step 3: Use the Point-Slope Form to Find the Equation of the Perpendicular Line

Using the point-slope form with the slope (2) and the point ((2, -1)):

y - (-1)  2(x - 2)y   1  2x - 4y  2x - 5

Converting to standard form:

2x - y - 5  0

Conclusion

The equation of the line perpendicular to (2x 4y 3 0) and passing through the point ((2, -1)) is (y 2x - 5) or (2x - y - 5 0).

Summary

To find the equation of a line that is perpendicular to a given line and passes through a specific point, we first need to determine the slope of the given line. Then, we find the slope of the perpendicular line (which is the negative reciprocal of the given slope). Using the point-slope form or standard form, we can derive the equation of the perpendicular line.

Conclusion

Mastering the process of finding equations of perpendicular lines is essential for various applications in mathematics, including geometry and calculus. Practice and understanding these steps will significantly enhance your ability to solve such problems efficiently.