Equation Of A Perpendicular Line Through A Point

Hey guys! Today, we're diving deep into a classic coordinate geometry problem that's super common in math – finding the equation of a line when you know it's perpendicular to another line and passes through a specific point. This is a fundamental skill, and once you nail it, you'll see it pop up in all sorts of places, from calculus to physics. So, grab your notebooks, maybe a snack, and let's break down how to solve this step-by-step. We're going to take the problem: "The equation of line $k$ is $y=-10 x-8$. Perpendicular to line $k$ is line $\ell$, which passes through the point $(2,2)$. What is the equation of line $\ell$? Write the equation in slope-intercept form. Write the numbers in the equation as simplified fractions." and make it crystal clear.

Understanding Perpendicular Lines and Slope-Intercept Form

Before we jump into solving, let's get our heads around the key concepts. First up, perpendicular lines. What does it mean for two lines to be perpendicular? Basically, they intersect at a right angle, like the corner of a book or the hands of a clock at 3:00. In the world of coordinate geometry, this has a direct relationship with their slopes. If you have two lines, line 1 with slope $m_1$ and line 2 with slope $m_2$, and they are perpendicular, then the product of their slopes is -1. That is, $m_1 \times m_2 = -1$. This also means that the slope of one line is the negative reciprocal of the other. So, if line $k$ has a slope of $m_k$, the slope of a line $\ell$ perpendicular to it, let's call it $m_\ell$, will be $m_\ell = -1/m_k$. This is our golden ticket for finding the slope of the new line we're looking for. Keep this relationship handy, guys!

Now, let's talk about slope-intercept form. This is probably the most common way we write the equation of a line. It looks like this: $y = mx + b$. Here, '$m$' represents the slope of the line (which we just talked about!), and '$b$' represents the y-intercept – the point where the line crosses the y-axis. Our goal in this problem is to find the specific values for '$m$' and '$b$' that define line $\ell$. We're already given that the final answer should be in this nice, neat $y = mx + b$ format, with any numbers written as simplified fractions. This means we need to be extra careful with our calculations, especially when dealing with negatives and fractions, to make sure our final answer is as tidy as possible.

Step 1: Identify the Slope of the Given Line

Alright, let's get down to business with our specific problem. We are given line $k$ with the equation $y = -10x - 8$. This equation is already in slope-intercept form ($y = mx + b$). This is super convenient because we can immediately identify the slope of line $k$. In this equation, the coefficient of $x$ is $-10$. So, the slope of line $k$, which we can denote as $m_k$, is -10. This is our starting point. We know the slope of the line we're not looking for. This number is crucial because it will help us find the slope of the line we are looking for.

Think about it: if you're trying to draw a line that's perfectly perpendicular to another one, you need to know the 'steepness' and 'direction' of the original line. The slope tells us exactly that. A slope of -10 means that for every 1 unit you move to the right on the x-axis, the line goes down by 10 units on the y-axis. It's a pretty steep downward slope! Recognizing this slope is the first, and arguably the easiest, step in solving this problem. Don't underestimate the power of this first piece of information; it unlocks the next step.

Step 2: Calculate the Slope of the Perpendicular Line

Now that we know the slope of line $k$ is $m_k = -10$, we can use our rule for perpendicular lines to find the slope of line $\ell$. Remember, the slopes of perpendicular lines are negative reciprocals of each other. This means we need to do two things to $m_k$: take its reciprocal and change its sign. So, for line $\ell$, its slope, $m_\ell$, will be:

$m_\ell = -\frac{1}{m_k}$

Plugging in our value for $m_k$:

$m_\ell = -\frac{1}{-10}$

When you have a negative divided by a negative, it becomes a positive. So,

$m_\ell = \frac{1}{10}$

Boom! We've just found the slope of line $\ell$. It's $\frac{1}{10}$. This is a much gentler upward slope compared to the steep downward slope of line $k$. It means for every 10 units you move to the right on the x-axis, the line goes up by 1 unit on the y-axis. This step is where many people might make a small error, perhaps forgetting to change the sign or messing up the fraction. Always double-check: take the original slope, flip it (reciprocal), and change the sign. If the original slope was positive, the perpendicular slope will be negative, and vice versa. If the original slope was a fraction, flip the numerator and denominator. If it was a whole number, put it over 1 and then flip. For our case, $-10$ becomes $-1/(-10)$, which simplifies to $1/10$. Keep this fraction $\frac{1}{10}$ handy, as it's the '$m$' in our $y=mx+b$ equation for line $\ell$.

Step 3: Use the Point-Slope Form or Slope-Intercept Form

We've got the slope of line $\ell$ ($m_\ell = \frac{1}{10}$), and we know it passes through the point $(2,2)$. Now we need to find the y-intercept, '$b$', to complete the equation $y = mx + b$. There are a couple of ways to do this, but we'll explore the most common methods.

Method 1: Using the Slope-Intercept Form Directly

We know $y = mx + b$. We can substitute the slope we found ($m = \frac{1}{10}$) and the coordinates of the point it passes through ($x=2$, $y=2$) into this equation and solve for $b$.

$2 = (\frac{1}{10})(2) + b$

First, multiply the slope by the x-coordinate:

$2 = \frac{2}{10} + b$

Simplify the fraction $\frac{2}{10}$:

$2 = \frac{1}{5} + b$

Now, to isolate $b$, subtract $\frac{1}{5}$ from both sides of the equation:

$2 - \frac{1}{5} = b$

To subtract these, we need a common denominator. We can write $2$ as $\frac{10}{5}$:

$\frac{10}{5} - \frac{1}{5} = b$

$\frac{9}{5} = b$

So, the y-intercept '$b$' is $\frac{9}{5}$.

Method 2: Using the Point-Slope Form

Another powerful tool is the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$. Here, '$m$' is the slope, and $(x_1, y_1)$ is a point on the line. We already have both: $m = \frac{1}{10}$ and $(x_1, y_1) = (2,2)$. Let's plug these values in:

$y - 2 = \frac{1}{10}(x - 2)$

This equation represents our line $\ell$. However, the question asks for the equation in slope-intercept form ($y = mx + b$). So, we need to rearrange this equation. First, distribute the $\frac{1}{10}$ on the right side:

$y - 2 = \frac{1}{10}x - \frac{1}{10}(2)$

$y - 2 = \frac{1}{10}x - \frac{2}{10}$

Simplify the fraction $\frac{2}{10}$:

$y - 2 = \frac{1}{10}x - \frac{1}{5}$

Now, to get $y$ by itself, add 2 to both sides of the equation:

$y = \frac{1}{10}x - \frac{1}{5} + 2$

Again, we need to combine the constant terms. To add $- \frac{1}{5}$ and $2$, we write $2$ as $\frac{10}{5}$:

$y = \frac{1}{10}x - \frac{1}{5} + \frac{10}{5}$

$y = \frac{1}{10}x + \frac{9}{5}$

Both methods give us the same y-intercept, $b = \frac{9}{5}$. This consistency is a good sign that we're on the right track, guys! You can use whichever method feels more comfortable to you.

Step 4: Write the Final Equation in Slope-Intercept Form

We've done all the heavy lifting! We have the slope ($m = \frac{1}{10}$) and the y-intercept ($b = \frac{9}{5}$) for our line $\ell$. The final step is to plug these values back into the slope-intercept form, $y = mx + b$.

So, the equation of line $\ell$ is:

$y = \frac{1}{10}x + \frac{9}{5}$

We also need to ensure that the numbers in the equation are written as simplified fractions, which they already are! $\frac{1}{10}$ cannot be simplified further, and neither can $\frac{9}{5}$. So, we don't need to do any extra simplification steps here. If we had ended up with, say, $\frac{2}{20}$ for the slope, we would have had to simplify it to $\frac{1}{10}$. It's always important to check for simplification at the end.

Verification: Does the line pass through (2,2)?

It's always a good practice to check our work. Let's plug the point $(2,2)$ into our final equation $y = \frac{1}{10}x + \frac{9}{5}$ and see if it holds true.

Substitute $x=2$ and $y=2$:

$2 = \frac{1}{10}(2) + \frac{9}{5}$

$2 = \frac{2}{10} + \frac{9}{5}$

Simplify $\frac{2}{10}$ to $\frac{1}{5}$:

$2 = \frac{1}{5} + \frac{9}{5}$

Now, add the fractions on the right side:

$2 = \frac{1+9}{5}$

$2 = \frac{10}{5}$

$2 = 2$

It works! The equation holds true for the point $(2,2)$. This confirms that our equation for line $\ell$ is correct.

Key Takeaways and Practice

So, to recap the entire process:

1. Identify the slope of the given line ($m_k$): From $y = -10x - 8$, we found $m_k = -10$.
2. Calculate the slope of the perpendicular line ($m_\ell$): The negative reciprocal of $-10$ is $\frac{1}{10}$.
3. Use the slope and the given point to find the y-intercept ($b$): We used $y = mx + b$ (or point-slope form) with $m=\frac{1}{10}$ and $(x,y)=(2,2)$ to find $b = \frac{9}{5}$.
4. Write the final equation: Substitute $m$ and $b$ into $y = mx + b$ to get $y = \frac{1}{10}x + \frac{9}{5}$.

Practice makes perfect, guys! Try this with different lines and points. What if the given line was horizontal ($y=c$)? Its slope is 0. A perpendicular line would be vertical ($x=c$), which has an undefined slope. Our formula $m_1 imes m_2 = -1$ doesn't directly apply there, but the concept of perpendicularity does. What if the point had negative coordinates? Just be careful with your signs. The core steps remain the same.

This problem is a fantastic way to solidify your understanding of linear equations and their properties. Keep practicing, and you'll be solving these in your sleep! Let me know if you guys have any questions. Happy graphing!