Perpendicular To Line Through Point

Finding the Line Perpendicular to a Given Line Through a Specific Point

Finding the equation of a line perpendicular to another line and passing through a given point is a fundamental concept in coordinate geometry. This process involves understanding the relationship between slopes of perpendicular lines and applying the point-slope form of a linear equation. This comprehensive guide will walk you through the process step-by-step, covering various scenarios and providing ample examples to solidify your understanding. We'll explore both algebraic and geometric interpretations, ensuring a thorough grasp of this important mathematical concept.

Understanding Slopes and Perpendicular Lines

Before diving into the process, let's refresh our understanding of slopes and their relationship in perpendicular lines. The slope (often denoted as 'm') of a line represents its steepness. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line. Mathematically:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are two points on the line.

Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that if one line has a slope of 'm', a line perpendicular to it will have a slope of '-1/m'. This relationship is crucial for finding the equation of a perpendicular line. There's one exception: vertical and horizontal lines are perpendicular, even though the concept of slope doesn't directly apply to vertical lines (their slope is considered undefined).

Step-by-Step Process: Finding the Perpendicular Line

Let's outline the steps involved in finding the equation of a line perpendicular to a given line and passing through a specific point.

1. Determine the Slope of the Given Line:

First, you need the equation of the given line. This equation can be in various forms (slope-intercept form: y = mx + b; standard form: Ax + By = C; or point-slope form: y - y₁ = m(x - x₁)). Regardless of the form, you need to find the slope 'm' of the given line. If the equation is in slope-intercept form (y = mx + b), the slope 'm' is readily apparent. If it's in standard form (Ax + By = C), you can rearrange it to slope-intercept form to find the slope. If you have two points on the line, use the slope formula mentioned earlier.

2. Find the Slope of the Perpendicular Line:

Once you have the slope 'm' of the given line, find the negative reciprocal of this slope. This will be the slope of the line perpendicular to the given line. Let's denote this new slope as 'm⊥' (m-perpendicular). The formula is:

m⊥ = -1/m

3. Use the Point-Slope Form:

Now, you need a point through which the perpendicular line passes. Let's say this point has coordinates (x₁, y₁). Use the point-slope form of a linear equation to construct the equation of the perpendicular line:

y - y₁ = m⊥(x - x₁)

where:

• y and x are the variables representing any point on the perpendicular line.
• y₁ and x₁ are the coordinates of the given point.
• m⊥ is the slope of the perpendicular line calculated in step 2.

4. Simplify the Equation (Optional):

Finally, simplify the equation obtained in step 3 into a more convenient form, such as the slope-intercept form (y = mx + b) or the standard form (Ax + By = C). This makes the equation easier to interpret and use in further calculations.

Examples: Illustrating the Process

Let's work through a few examples to illustrate the process clearly.

Example 1: Given Line in Slope-Intercept Form

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

1. Slope of the given line: The slope 'm' is 2.
2. Slope of the perpendicular line: m⊥ = -1/2
3. Point-slope form: y - 1 = -1/2(x - 4)
4. Simplified form: y = -1/2x + 3

Therefore, the equation of the perpendicular line is y = -1/2x + 3.

Example 2: Given Line in Standard Form

Find the equation of the line perpendicular to 3x + 4y = 12 and passing through the point (-2, 5).

1. Slope of the given line: Rearrange the equation to slope-intercept form: 4y = -3x + 12 => y = (-3/4)x + 3. The slope 'm' is -3/4.
2. Slope of the perpendicular line: m⊥ = -1/(-3/4) = 4/3
3. Point-slope form: y - 5 = (4/3)(x - (-2)) => y - 5 = (4/3)(x + 2)
4. Simplified form: y = (4/3)x + 13/3

Therefore, the equation of the perpendicular line is y = (4/3)x + 13/3.

Example 3: Handling a Vertical Line

Find the equation of the line perpendicular to x = 5 (a vertical line) and passing through the point (2, 3).

A vertical line has an undefined slope. A line perpendicular to a vertical line is a horizontal line. Horizontal lines have a slope of 0.

1. Slope of the perpendicular line: m⊥ = 0
2. Point-slope form: y - 3 = 0(x - 2)
3. Simplified form: y = 3

Therefore, the equation of the perpendicular line is y = 3.

Geometric Interpretation

The geometric interpretation helps visualize the concept. Perpendicular lines intersect at a right angle (90 degrees). The negative reciprocal relationship of slopes ensures this right-angle intersection. Imagine drawing the two lines on a coordinate plane. The steeper the original line, the less steep the perpendicular line will be, and vice-versa. The intersection point is the point specified in the problem.

Advanced Scenarios and Considerations

While the steps outlined above cover most common scenarios, let's consider some more complex situations:

• Parallel Lines: If two lines are parallel, they have the same slope. Finding a line perpendicular to a parallel line pair involves finding the perpendicular to one of the lines and then using that slope for the other.

• Lines Defined by Two Points: If the given line is defined by only two points, first calculate its slope using the slope formula, then proceed with the steps outlined above.

• Systems of Equations: In more advanced applications, you might encounter problems requiring solving systems of equations to find the intersection point of the given line and its perpendicular line.

Frequently Asked Questions (FAQ)

• Q: What if the slope of the given line is 0?

  A: If the slope of the given line is 0 (meaning it's a horizontal line), the perpendicular line will be a vertical line with an undefined slope. Its equation will be of the form x = constant, where the constant is the x-coordinate of the given point.

• Q: Can I use other forms of linear equations besides point-slope form?

  A: Yes, you can use the slope-intercept form (y = mx + b) or the standard form (Ax + By = C), but the point-slope form is generally the most convenient for this specific problem.

• Q: What if the given point lies on the given line?

  A: If the given point lies on the given line, then there is no unique perpendicular line through that point. Infinite perpendicular lines could pass through that point.

• Q: How can I check my answer?

  A: After finding the equation of the perpendicular line, substitute the coordinates of the given point into the equation to ensure it satisfies the equation. Also, check that the product of the slopes of the two lines is -1 (excluding cases with vertical or horizontal lines).

Conclusion

Finding the equation of a line perpendicular to a given line and passing through a specific point is a crucial skill in coordinate geometry. By understanding the relationship between slopes of perpendicular lines and applying the point-slope form of a linear equation, you can effectively solve a wide range of problems. Remember to always double-check your calculations and consider the geometric interpretation to enhance your understanding and problem-solving abilities. Practice with different examples and scenarios to solidify your understanding of this fundamental concept.