Understanding Perpendicular Lines Through the Origin

The concept of finding the equation of a line that is perpendicular to another line and passes through the origin is a fundamental topic in algebra and geometry. In this article, we will explore the process of finding such a line, using a specific example: y 1/5x.

Determining the Slope of the Given Line

Let's begin by identifying the slope of the given line. The equation y 1/5x is in the slope-intercept form y mx b, where:

m represents the slope of the line. b represents the y-intercept.

In the given equation, y 1/5x, the slope m_1 is 1/5. If you try to express this in the form 5y x or x - 5y 0, it becomes evident that the slope remains 1/5.

Steps to Find the Perpendicular Line

To find the equation of a line that is perpendicular to the given line and passes through the origin, follow these steps:

1. Determine the Slope of the Given Line

The slope of the given line is 1/5.

2. Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope m_2 of the perpendicular line can be calculated as follows:

m_2 -1/m_1 -1/(1/5) -5

3. Use the Point-Slope Form to Write the Equation of the Perpendicular Line

Since the line passes through the origin (0, 0), the point-slope form of the equation of the line can be written as:

y - y_1 m(x - x_1)

Substituting m -5 and (x_1, y_1) (0, 0):

y - 0 -5(x - 0)

Simplifying this, we get:

y -5x

Therefore, the equation of the line that is perpendicular to y 1/5x and passes through the origin is:

y -5x

Verification of the Solution

To ensure the correctness of our solution, let’s verify using the formula for perpendicular lines:

m_1m_2 -1

With m_1 1/5 and m_2 -5:

(1/5) * (-5) -1

This confirms that the slope of the lines are indeed perpendicular.

Additional Methods and Formulations

There are two additional methods to derive the equation of the perpendicular line:

Method 1: Using the General Equation Form

The general form of a line is y mx c. For the perpendicular line passing through the origin:

0 -5(0) c

This simplifies to:

c 0

Hence, the equation of the line is:

y -5x

Method 2: Using the Perpendicular Slope and Origin Point

The equation of the line in the slope-intercept form is:

y mx

Substituting the slope m -5, we get:

y -5x

Both methods lead us to the same conclusion: y -5x.

Conclusion

In conclusion, finding the equation of a line that is perpendicular to another line and passes through the origin involves identifying the slope of the original line, calculating the negative reciprocal to find the slope of the perpendicular line, and using the point-slope form to write the equation. For the given line y 1/5x, the equation of the line that is perpendicular and passes through the origin is y -5x.