Rotating Axes: Eliminating Xy-Term & Standard Form
Hey math enthusiasts! Let's dive into a cool concept: rotating axes to get rid of that pesky xy-term in equations. This technique is super useful for simplifying equations and understanding the shapes they represent. We'll walk through the process, making it easy to grasp. We'll be focusing on the equation xy + 1 = 0 and transforming it into its standard form.
Understanding the Need to Rotate
So, why do we even bother rotating axes? Well, equations with an xy-term, like the one we've got, are a bit tricky to visualize directly. They often represent conic sections (ellipses, hyperbolas, parabolas) that are tilted or rotated in the coordinate plane. The xy-term causes this rotation. By rotating the axes, we can align the conic section with the new axes, making the equation simpler and easier to work with. Think of it like straightening a crooked picture frame – we're just making things line up nicely.
Imagine the equation xy + 1 = 0. Without any fancy transformations, it's hard to tell what kind of shape we're dealing with. The xy-term hints at a hyperbola, but it's rotated. Rotating the axes will straighten out this hyperbola, making it much easier to identify its key features (vertices, foci, asymptotes, etc.) and sketch its graph. This process essentially undoes the rotation, presenting the conic section in its most straightforward orientation. It's like having a map of a city, and then rotating the map to better align with the direction you are facing. This way, you understand the city layout quickly.
To make this transformation, we will use the rotation formulas. This involves finding an angle of rotation (θ) and applying the transformation formulas:
- x = x'cos(θ) - y'sin(θ)
- y = x'sin(θ) + y'cos(θ)
We will use these formulas to replace x and y in the original equation and derive a new equation in terms of x' and y'. This will eliminate the xy-term. This also enables us to understand the underlying geometry and properties more easily. Rotating axes is a fundamental skill in analytic geometry and finds applications in various fields.
The Rotation Formulas: Your Transformation Toolkit
Alright, let's get into the nitty-gritty of the rotation formulas. These formulas are the heart of the transformation process. They allow us to relate the original x and y coordinates to the new x' and y' coordinates after the rotation. Here’s a breakdown:
- x = x'cos(θ) - y'sin(θ): This formula tells us how to express the original x-coordinate in terms of the new coordinates (x', y') and the angle of rotation (θ). Essentially, it's saying the original x is a combination of the new x' and y' values, scaled by the cosine and sine of the rotation angle.
- y = x'sin(θ) + y'cos(θ): Similarly, this formula expresses the original y-coordinate in terms of the new coordinates and the angle θ. It’s the counterpart to the x transformation, ensuring we account for the rotation in both directions. The new coordinates are related to the original coordinates by trigonometric functions (sin and cos) of the angle of rotation, reflecting a change in perspective.
These formulas are derived from trigonometric principles. Picture a point in the plane, and then rotate the axes around the origin. The point itself doesn’t move, but its coordinates change relative to the rotated axes. The formulas are constructed based on the concept that a point's coordinates can be expressed differently depending on the orientation of the coordinate system. You can picture it as the point remaining stationary while the coordinate system spins.
To eliminate the xy-term, we'll need to find the right angle of rotation (θ). This angle is determined by the coefficients of the x², xy, and y² terms in the original equation. We'll use the formula: cot(2θ) = (A - C) / B, where A, B, and C are the coefficients of x², xy, and y², respectively. In our case, since there is no x² or y² terms, the formula simplifies. Once we find θ, we can plug it into the rotation formulas and simplify the original equation.
Finding the Angle of Rotation (θ)
Now, let's find the angle of rotation (θ). This is a critical step because it dictates how much we're rotating the axes. The goal is to choose an angle that eliminates the xy-term. It's like finding the perfect angle to take a picture so that the subject is perfectly framed and the background clutter disappears. The formula we use is derived from the coefficients of the terms in the original equation.
The general formula for finding the angle of rotation, derived to eliminate the xy-term, is: cot(2θ) = (A - C) / B, where:
- A is the coefficient of the x² term
- B is the coefficient of the xy term
- C is the coefficient of the y² term
In our equation, xy + 1 = 0, we can rewrite it as 0x² + 1xy + 0y² + 1 = 0. This means:
- A = 0
- B = 1
- C = 0
Plugging these values into the formula, we get: cot(2θ) = (0 - 0) / 1 = 0. Therefore, cot(2θ) = 0. This implies that 2θ = 90 degrees or π/2 radians. Thus, θ = 45 degrees or π/4 radians. This means we'll rotate the axes by 45 degrees. Knowing the correct rotation angle is like knowing which way to turn the steering wheel to navigate a turn.
The key takeaway here is how the coefficients of the quadratic terms in the equation dictate the necessary rotation. This formula is derived from trigonometric identities and ensures the xy-term is eliminated through a well-defined rotation. Finding the correct θ is crucial; it’s the key that unlocks the simplification of the equation and reveals its true nature.
Applying the Rotation: Transforming the Equation
Now, let's apply the rotation formulas using the angle of rotation we just found (θ = 45 degrees or π/4 radians). This means we'll substitute x and y in the original equation with expressions involving x', y', and the trigonometric functions of 45 degrees. The process will transform the equation into a form without an xy-term.
Remember our rotation formulas:
- x = x'cos(θ) - y'sin(θ)
- y = x'sin(θ) + y'cos(θ)
Since θ = 45 degrees, we have cos(45°) = √2 / 2 and sin(45°) = √2 / 2. Plugging these values, the formulas become:
- x = x'(√2 / 2) - y'(√2 / 2)
- y = x'(√2 / 2) + y'(√2 / 2)
Now, let's substitute these into our original equation, xy + 1 = 0:
(x'(√2 / 2) - y'(√2 / 2)) * (x'(√2 / 2) + y'(√2 / 2)) + 1 = 0
Expanding this, we get:
(x'²/2 - y'²/2) + 1 = 0
Simplifying, we have: x'²/2 - y'²/2 = -1
Multiplying both sides by -2: -x'² + y'² = 2. Finally: (y'²/2) - (x'²/2) = 1. This is the new equation.
This new equation is in standard form. Notice the absence of the x'y'-term! This tells us we have a hyperbola. The process might seem a bit involved, but it's a systematic way to eliminate the xy-term and simplify the equation.
Writing the Equation in Standard Form: The Grand Finale
We've done the hard work, so let's get that final equation in standard form. Remember, our goal is to rewrite the equation in a way that clearly shows the type of conic section and its key features. Standard forms help us easily identify and graph the equation.
We ended up with the equation: (y'²/2) - (x'²/2) = 1. Let's analyze this equation:
- It has an x'² term and a y'² term, but no x'y' term. This confirms that our rotation successfully eliminated the xy-term.
- The coefficients of x'² and y'² are different and have opposite signs, which tells us that the equation represents a hyperbola. If the signs were the same, it would be an ellipse (or a circle, if the coefficients were equal). If one term was squared and the other wasn't, it would be a parabola.
- We can rewrite the equation as (y'²/2) - (x'²/2) = 1, to make it even clearer. This is now in standard form for a hyperbola centered at the origin of the rotated coordinate system. It shows that the hyperbola opens up and down along the y'-axis.
So, the standard form of the equation after rotating the axes is (y'²/2) - (x'²/2) = 1. This form allows us to quickly identify the type of conic section and easily sketch its graph. Understanding standard forms is crucial for analyzing and working with conic sections. It tells you the center, orientation, and other key information about the shape. From here, you could easily find the vertices, foci, and asymptotes of the hyperbola.
Conclusion: The Power of Axis Rotation
And there you have it, guys! We've successfully rotated the axes to eliminate the xy-term and transformed the equation xy + 1 = 0 into its standard form, revealing the underlying hyperbola. This is a powerful technique for simplifying equations and understanding their geometric representations.
Key Takeaways:
- Rotating axes simplifies equations with an xy-term.
- The angle of rotation is determined using the coefficients of the equation.
- The rotation formulas are the tools for the transformation.
- Standard form helps in identifying and graphing the conic section.
Keep practicing, and you'll get the hang of it! Math can be fun, and this is a great example of how we can manipulate equations to reveal their secrets. Keep exploring and happy calculating!