Lines Through Origin At 60° To Y-Axis: Combined Equation

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Hey guys! Let's dive into a cool math problem today: finding the combined equation of lines that pass through the origin and make a 60-degree angle with the Y-axis. Sounds a bit tricky, right? But don't worry, we'll break it down step by step so it's super easy to understand. We'll cover everything from the basic concepts to the final equation, making sure you've got a solid grasp on how to tackle these kinds of problems. So, grab your calculators and let's get started!

Understanding the Basics

Before we jump into solving the problem, let's quickly refresh some fundamental concepts. This will help us build a strong foundation and make the entire process smoother. Think of it as setting the stage before the main performance – we want to make sure everything is in place for a stellar show!

What is the Origin?

First off, what exactly is the origin? In a two-dimensional coordinate system (like our good old x-y plane), the origin is the point where the x-axis and y-axis intersect. It's that sweet spot right in the middle, represented by the coordinates (0, 0). All our lines will be passing through this central point, so keep it in mind!

Lines and Angles

Now, let’s talk about lines and angles. A line’s angle with an axis tells us how much it's tilted away from that axis. In this case, we're dealing with lines that make a 60-degree angle with the Y-axis. Imagine the Y-axis as standing straight up, and our lines are leaning away from it at 60 degrees. This angle is crucial because it helps us determine the slope of the lines, which is a key ingredient in finding their equations.

The Equation of a Line

Speaking of equations, remember the general equation of a line? There are a few forms, but the one we'll use most here is the slope-intercept form: y = mx + c. Here,

  • y and x are the coordinates of any point on the line.
  • m is the slope of the line (how steep it is).
  • c is the y-intercept (where the line crosses the y-axis).

Since our lines pass through the origin (0, 0), the y-intercept c will be 0. This simplifies our equation to y = mx, which is super handy!

Combined Equation

Lastly, what do we mean by a "combined equation"? When we have multiple lines, each with its own equation, the combined equation is a single equation that represents all of those lines together. It's like writing a single story that includes all the characters and plotlines from individual stories. We'll achieve this by multiplying the individual line equations, setting the product equal to zero. This works because if any one of the individual equations is zero, the entire product becomes zero, satisfying the condition for all lines.

Why This Matters

Understanding these basics isn't just about solving this specific problem. It's about building a solid mathematical foundation. Knowing these concepts will help you tackle a wide range of problems involving lines, angles, and equations. Plus, it's pretty cool to see how different mathematical ideas fit together, right? So, with these building blocks in place, let's move on to the nitty-gritty of our problem!

Setting Up the Problem

Alright, now that we've brushed up on the basics, let’s dive into the specifics of our problem. We need to find the combined equation of lines that pass through the origin and make a 60-degree angle with the Y-axis. Let’s break down what this means and how we can start setting up our solution.

Visualizing the Lines

The first step in solving any geometry problem is to visualize what’s going on. Imagine the coordinate plane with the x and y axes. The origin is right in the center, at (0, 0). Now, picture a line passing through this origin. This line forms an angle with the Y-axis. Our problem specifies that this angle is 60 degrees. But here’s the catch: there isn’t just one line that does this. There are actually two lines that pass through the origin and make a 60-degree angle with the Y-axis – one on each side of the Y-axis. Think of it like the hands of a clock, where one hand is 60 degrees to the right and the other is 60 degrees to the left of the 12 o'clock position.

Determining the Angles with the X-axis

To find the equations of these lines, it's more convenient to work with the angles they make with the X-axis. Why? Because the slope of a line is directly related to the tangent of the angle it makes with the X-axis. So, we need to figure out those angles. Remember that the X and Y axes are perpendicular, meaning they form a 90-degree angle. If a line makes a 60-degree angle with the Y-axis, it will make a different angle with the X-axis. To find this angle, we can use some simple geometry. If we have a line that forms a 60-degree angle with the positive Y-axis, the angle it forms with the positive X-axis will be 90 degrees + 60 degrees = 150 degrees.

However, we also have a line on the other side of the Y-axis. This line forms a 60-degree angle with the negative Y-axis. So, the angle this line makes with the positive X-axis will be 90 degrees - 60 degrees = 30 degrees. So, we now know that we have two lines: one making a 30-degree angle with the X-axis and the other making a 150-degree angle with the X-axis. These angles are crucial for finding the slopes of our lines.

Finding the Slopes

Now comes the fun part: calculating the slopes. The slope (m) of a line is equal to the tangent of the angle (θ) it makes with the X-axis. Mathematically, this is expressed as m = tan(θ). We have two angles, so we'll have two slopes. For the line making a 30-degree angle with the X-axis, the slope (m1) is:

m1 = tan(30°) = 1/√3

For the line making a 150-degree angle with the X-axis, the slope (m2) is:

m2 = tan(150°) = -1/√3

Notice that the slopes have opposite signs. This makes sense because one line is sloping upwards (positive slope) and the other is sloping downwards (negative slope).

Setting Up the Equations

We're almost there! Now that we have the slopes, we can write the equations of the individual lines. Remember that the equation of a line passing through the origin is y = mx. So, for our two lines, the equations are: For the line with a slope of 1/√3:

y = (1/√3)x

Which can be rewritten as:

√3y = x

Or:

x - √3y = 0

For the line with a slope of -1/√3:

y = (-1/√3)x

Which can be rewritten as:

√3y = -x

Or:

x + √3y = 0

So, we’ve successfully set up the equations for the two lines. Now, the final step is to combine these equations into a single equation. This is where the magic happens, and we see how individual pieces come together to form a complete solution.

Calculating the Combined Equation

Okay, guys, we've reached the final stretch! We have the individual equations for the two lines, and now we need to find their combined equation. Remember, the combined equation is a single equation that represents both lines. We achieve this by multiplying the individual equations together. Let's see how it's done.

Multiplying the Equations

We found the equations of the two lines to be:

  1. x - √3y = 0
  2. x + √3y = 0

To find the combined equation, we simply multiply these two equations: (x - √3y) * (x + √3y) = 0

Now, we need to expand this product. Notice that this is in the form of (a - b)(a + b), which is a difference of squares. Remember the formula: (a - b)(a + b) = a² - b². This will make our calculation much simpler.

Expanding the Product

Using the difference of squares formula, we get:

x² - (√3y)² = 0

Now, let's simplify the second term: (√3y)² = (√3)² * y² = 3y²

So, our equation becomes:

x² - 3y² = 0

And that's it! This is the combined equation of the lines passing through the origin and making a 60-degree angle with the Y-axis.

Checking Our Work

It’s always a good idea to double-check our answer to make sure everything makes sense. The combined equation x² - 3y² = 0 represents two lines passing through the origin. The absence of constant terms confirms this. The presence of both x² and y² terms indicates that we are dealing with a pair of straight lines, which is exactly what we expected. Also, if we were to solve this equation for y in terms of x, we would get two solutions corresponding to the individual lines we found earlier. This gives us confidence that our answer is correct.

Significance of the Result

This combined equation is not just a solution to a specific problem; it provides a general way to represent pairs of lines passing through the origin. This technique is used in various areas of mathematics and physics. For instance, it can help in analyzing geometric shapes, understanding the behavior of light rays, and solving problems in mechanics. Understanding how to derive and interpret combined equations is a valuable skill in many fields.

Conclusion

So, there you have it! We’ve successfully found the combined equation of the lines passing through the origin and making a 60-degree angle with the Y-axis. We started by understanding the basic concepts, visualized the problem, calculated the slopes, derived the individual line equations, and finally, combined them into a single equation. It might seem like a lot of steps, but each one is crucial for solving the problem accurately. Remember, math is like building a house – you need a solid foundation and a step-by-step approach to get the final structure right.

Key Takeaways

Let’s quickly recap the key points from our journey today:

  • Visualizing the problem is the first and often most important step. It helps you understand what’s going on and guides your approach.
  • Understanding the relationship between angles and slopes (m = tan(θ)) is fundamental in coordinate geometry.
  • The equation of a line passing through the origin is y = mx, which simplifies calculations.
  • The combined equation of multiple lines is found by multiplying their individual equations.
  • Using algebraic identities like the difference of squares can greatly simplify your calculations.
  • Checking your work ensures that your answer makes sense and is likely correct.

Final Thoughts

I hope you guys found this explanation helpful and easy to follow. Math can seem intimidating at first, but breaking it down into smaller, manageable steps makes it much more approachable. Practice is key, so try applying these concepts to similar problems. The more you practice, the more confident you'll become. And remember, every problem you solve is a step forward in your mathematical journey. Keep up the great work, and I’ll see you in the next math adventure!