Finding The Equation Of A Perpendicular Line A Comprehensive Guide

Finding the Equation of a Perpendicular Line Passing Through a Point

Hey guys! Today, we're diving into a common problem in coordinate geometry: finding the equation of a line that's perpendicular to a given line and passes through a specific point. This is a fundamental concept, and mastering it will definitely boost your math skills. Let's break it down step by step.

Understanding the Basics

Before we jump into solving the problem, let's quickly recap some essential concepts. Remember, the equation of a line can be expressed in several forms, but the most common ones we'll use here are the slope-intercept form ( $y = mx + c$ ) and the point-slope form ( $y - y_1 = m(x - x_1)$ ). The slope (m) tells us how steep the line is, and the y-intercept (c) is where the line crosses the y-axis. The point-slope form is super handy when we know a point on the line ( $x_1$ , $y_1$ ) and its slope.

Now, when we talk about perpendicular lines, there's a crucial relationship between their slopes. If two lines are perpendicular, the product of their slopes is -1. In simpler terms, if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This is the key to solving our problem. Got it? Great!

Key Concepts to Remember

Slope-intercept form: The equation of a line in the form ${y = mx + c}$ , where m is the slope and c is the y-intercept. This form is excellent for visualizing the line's characteristics directly from the equation.
Point-slope form: The equation of a line in the form ${y - y_1 = m(x - x_1)}$ , where ${(x_1, y_1)}$ is a point on the line and m is the slope. This form is particularly useful when you have a point and the slope and want to find the equation.
Perpendicular lines: Two lines are perpendicular if the product of their slopes is -1. If line 1 has a slope of ${m_1}$ and line 2 has a slope of ${m_2}$ , then for the lines to be perpendicular, ${m_1 \times m_2 = -1}$ . This is a fundamental concept in coordinate geometry.

Understanding these basics thoroughly will make the process of finding the equation of a perpendicular line much smoother. With these concepts in mind, let's move on to the problem at hand.

Problem Statement: Finding the Perpendicular Line

Okay, let's dive into the problem we're tackling today. We're given a line, let's call it line g, with the equation $3x - y = 3P - Q$. We need to find the equation of another line that is perpendicular to line g and passes through a specific point, $(P, Q)$. This is a classic coordinate geometry problem, and we're going to break it down step by step so it’s super clear. Remember, the goal is to find the equation of a new line that satisfies two conditions: it must be perpendicular to the given line, and it must contain the given point. Got it? Let’s get started.

Breaking Down the Problem

To make this problem easier, we need to approach it methodically. First, we'll determine the slope of the given line g. Remember, the slope tells us how steep the line is. Then, using the relationship between the slopes of perpendicular lines, we'll find the slope of the line we're trying to find. Finally, we'll use the point-slope form of a line equation to write the equation of the new line, since we know a point it passes through and its slope.

This approach is like a roadmap. By breaking the problem into smaller, manageable steps, we can tackle each part confidently. It’s all about understanding the underlying principles and applying them systematically. So, let's follow this roadmap and start with the first step: finding the slope of the given line.

Step-by-Step Approach

Find the slope of the given line g: This involves rearranging the equation of line g into slope-intercept form. This is a crucial first step because we need to know the slope of the original line to determine the slope of the perpendicular line.
Determine the slope of the perpendicular line: Using the fact that the product of the slopes of perpendicular lines is -1, we can find the slope of the line we're looking for. This step connects the slope of the given line to the slope of our new line.
Use the point-slope form to find the equation: With the slope of the perpendicular line and the given point $(P, Q)$, we can write the equation of the line. This is where we bring everything together to form the final equation.

With this breakdown, the problem becomes much less daunting. We have a clear path to follow, and each step builds upon the previous one. Now, let's put this plan into action and start solving the problem.

Step 1: Finding the Slope of Line $g$

Alright, the first thing we need to do is find the slope of line $g$. The equation for line $g$ is given as $3x - y = 3P - Q$. To easily identify the slope, we need to rewrite this equation in the slope-intercept form, which is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. This form makes the slope jump right out at you! So, let’s get this equation into the right shape.

Rewriting the Equation

To transform $3x - y = 3P - Q$ into slope-intercept form, we need to isolate $y$ on one side of the equation. Here's how we do it:

Start with the original equation: $3x - y = 3P - Q$
Subtract $3x$ from both sides: $-y = -3x + 3P - Q$
Multiply both sides by -1: $y = 3x - 3P + Q$

Now, the equation is in the form $y = mx + c$. Can you spot the slope? It's the coefficient of $x$.

Identifying the Slope

Looking at our rearranged equation, $y = 3x - 3P + Q$, the slope, $m$, of line $g$ is clearly 3. This is a crucial piece of information. We now know how steep line $g$ is, and we can use this to figure out the slope of any line perpendicular to it.

Understanding this step is fundamental. The ability to manipulate equations into different forms to extract key information is a core skill in algebra and coordinate geometry. So, we've successfully found the slope of line $g$. What's next? Time to use this information to find the slope of the perpendicular line!

Step 2: Determining the Slope of the Perpendicular Line

Now that we've found the slope of line $g$, which is 3, we can figure out the slope of a line perpendicular to it. Remember the key fact: the slopes of perpendicular lines are negative reciprocals of each other. This means if one line has a slope of $m$, a line perpendicular to it will have a slope of $-1/m$. This is the golden rule we need to apply here.

Applying the Negative Reciprocal

Since the slope of line $g$ is 3, the slope of a line perpendicular to $g$ will be $-1/3$. That's it! We've found the slope of our new line. Let's call this slope $m_{\perp}$, so $m_{\perp} = -1/3$. This step is all about understanding and applying the relationship between slopes of perpendicular lines.

Why Negative Reciprocal Matters

The negative reciprocal relationship ensures that the lines intersect at a right angle (90 degrees). Think about it visually: a line with a positive slope goes upwards as you move from left to right, while a line with a negative slope goes downwards. The reciprocal part ensures the lines are steep enough to form a right angle, and the negative sign ensures they're going in opposite directions.

This concept is not just a mathematical trick; it has deep geometric significance. Understanding why this relationship holds true can help you visualize and solve many geometry problems. So, we now have the slope of the perpendicular line. What's the final piece of the puzzle? Figuring out the equation of the line using this slope and the given point.

Step 3: Finding the Equation of the Perpendicular Line

Okay, we're in the home stretch! We know the slope of the perpendicular line ($-1/3$) and we know it passes through the point $(P, Q)$. To find the equation of this line, we're going to use the point-slope form, which is a super useful tool in situations like this. The point-slope form of a line equation is: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Using the Point-Slope Form

In our case, we have:

$m = -1/3$ (the slope of the perpendicular line)$
$(x_1, y_1) = (P, Q)$ (the point the line passes through)$

Let's plug these values into the point-slope form:

y - Q = - rac{1}{3}(x - P)

This is a perfectly valid equation for our line, but sometimes it's helpful to rewrite it in slope-intercept form ($y = mx + c$) or standard form ($Ax + By = C$). Let's convert it to slope-intercept form to make it even clearer.

Converting to Slope-Intercept Form

To get to slope-intercept form, we need to isolate $y$. Here’s how:

Start with the point-slope equation: $y - Q = -rac{1}{3}(x - P)$
Distribute the $-1/3$: $y - Q = -rac{1}{3}x + rac{1}{3}P$
Add $Q$ to both sides: $y = -rac{1}{3}x + rac{1}{3}P + Q$

Now, we have the equation in slope-intercept form! The slope is $-1/3$ (as we already knew), and the y-intercept is $rac{1}{3}P + Q$. This equation tells us everything we need to know about the perpendicular line.

The Final Equation

So, the equation of the line that is perpendicular to $3x - y = 3P - Q$ and passes through the point $(P, Q)$ is:

y = - rac{1}{3}x + rac{1}{3}P + Q

Or, if you prefer the point-slope form:

y - Q = - rac{1}{3}(x - P)

We did it! We successfully found the equation of the perpendicular line. This problem involved several key steps, from finding the slope of the original line to using the point-slope form. Each step is crucial, and understanding the underlying concepts makes the whole process much smoother.

Alternative Approaches and Further Exploration

While we've solved the problem using the slope-intercept and point-slope forms, there are other ways to approach it. For example, we could have used the standard form of a line equation ($Ax + By = C$) to find the perpendicular line. The key takeaway is that there's often more than one path to the solution in mathematics, and exploring different methods can deepen your understanding.

Using Standard Form

Let's briefly explore how we could have used the standard form. The original line's equation, $3x - y = 3P - Q$, can be seen as already being in a form close to the standard form. A line perpendicular to $Ax + By = C$ will have the form $Bx - Ay = D$, where $D$ is a constant we need to determine. This method provides another perspective on solving these types of problems.

Further Practice

To really master this concept, try working through similar problems with different equations and points. Experiment with different forms of line equations and see which ones you find most intuitive. The more you practice, the more comfortable you'll become with coordinate geometry. Consider these variations for practice:

Find the equation of a line perpendicular to $x - y - P = 0$ passing through $(P, Q)$.
Find the equation of a line perpendicular to $x + y = Q - P$ passing through $(P, Q)$.
Find the equation of a line perpendicular to $3x + y = Q - 3P$ passing through $(P, Q)$.

By tackling these variations, you'll solidify your understanding and become more confident in your problem-solving abilities. Remember, mathematics is a journey of exploration and discovery.

Conclusion: Mastering Perpendicular Lines

So, there you have it, guys! We've successfully navigated the process of finding the equation of a line perpendicular to a given line and passing through a specific point. We started by understanding the basics, like slope-intercept and point-slope forms, and the crucial relationship between the slopes of perpendicular lines. Then, we broke down the problem into manageable steps, solved it methodically, and even explored alternative approaches.

The key takeaways here are: understanding the relationship between slopes of perpendicular lines, being comfortable with different forms of line equations, and having a systematic approach to problem-solving. These skills are not just useful for this specific type of problem but are valuable tools in your broader mathematical toolkit.

Remember, practice makes perfect. The more you work with these concepts, the more intuitive they'll become. So, keep practicing, keep exploring, and don't be afraid to tackle challenging problems. You've got this!