Finding the Equation of a Perpendicular Line

To find an equation of a line that is perpendicular to a given line, we need to first determine the slope of the original line. The process involves converting the given line into slope-intercept form and then finding the negative reciprocal of the slope. Let's explore these steps in detail.

Original Line: 5x 2y 12

First, convert the original line into slope-intercept form, which is represented by the equation y mx b where m is the slope and b is the y-intercept.

Step 1: Convert to Slope-Intercept Form

Start with the equation: 5x 2y 12 Subtract 5x from both sides: 2y -5x 12 Divide by 2 to solve for y: y -frac{5}{2}x 6

In this equation, the slope m -frac{5}{2}.

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is:

m -frac{1}{-frac{5}{2}} frac{2}{5}

The slope of the perpendicular line is (frac{2}{5}).

Step 3: Write the Equation of the Perpendicular Line

Given the slope of the perpendicular line and a point through which it passes, we can use the point-slope form to write the equation of the line. Let's choose the origin (0, 0).

y - 0 frac{2}{5}(x - 0)

Simplifying this, we get:

y frac{2}{5}x

Alternatively, you can express this in standard form:

2x - 5y 0

Thus, one equation of a line that is perpendicular to 5x 2y 12 is (y frac{2}{5}x).

Additional Examples

Example 1: 6x - 2y 12

Here, we follow similar steps:

Solve for y: 2y 6x - 12 Divide by 2: y 3x - 6 The slope, m, is 3. The slope of the perpendicular line is the negative reciprocal: -(frac{1}{3}). Using a point (0, 0), the equation is: y -(frac{1}{3})x.

Example 2: 6x - 2y 12 and through (-4, 3)

Convert the equation to slope-intercept form:

Solve for y: 2y 6x - 12 Divide by 2: y 3x - 6 The slope, m, is 3. The slope of the perpendicular line is (-(frac{1}{3}) Using a point (-4, 3), the equation in point-slope form is: y - 3 -(frac{1}{3})(x 4). Simplify: y - 3 -(frac{1}{3})x - (frac{4}{3}) Multiply by 3: 3y - 9 -x - 4 Final equation: x 3y 5

These steps illustrate the process of finding a line that is perpendicular to a given line, using either the slope-intercept form or the point-slope form. The key is to remember the negative reciprocal relationship between the slopes of perpendicular lines.