Problem 16 - Precalculus

Question

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$x y-8 x-4 y=0$$

Step-by-Step Solution

Answer

The standard form of the equation after rotating the axes through an angle of π/4 is \(X - Y = 0\). The graph of this equation is a hyperbola, which is centered at the origin and has asymptotes that are the lines X = Y and X = -Y. The original x and y axes are rotated by π/4.

Step 1: Re-write the Equation

Before starting with rotation of axes, let's reorder the equation a bit to get a clear view: \(xy -4y - 8x = 0\). Now we can proceed with the rotation.

Step 2: Rotate the Axes

The main step is to rotate the axes to eliminate the xy term. We can do this by replacing the variables x and y with functions of a new variable. In our case we will replace x and y by \(x = X cos(θ) - Y sin(θ)\) and \(y = X sin(θ) + Y cos(θ)\) respectively, where X and Y are coordinates of new system and θ is the angle of rotation. Upon substituting: \(X Y cos^2(θ) - X Y sin^2(θ) - 4 X sin(θ) - 4 Y cos(θ) - 8 X cos(θ) + 8 Y sin(θ) = 0\). In order for the xy term to disappear, we must set the coefficient in front of the XY term in the equation equal to zero: This gives us the trigonometric equation \(cos^2(θ) - sin^2(θ) = 0\), which have solutions at θ=π/4 and θ=3π/4. For simplicity, let's choose θ=π/4.

Step 3: Substitute the Angle and Simplify

By substituting θ=π/4 into the equation, X and Y becomes equal. Hence, we will get the new equation without the xy term: \(- 2X + 2Y = 0\) or, equivalently, \(X - Y = 0\). They represent the hyperbola rotated through an angle of π/4 in terms of the new variables X and Y.

Step 4: Plotting the result

In this step, we draw the hyperbola given by the equation \(X - Y = 0\) and then we draw the original x and y axes as rotated by π/4. This gives us a complete sketch of the graph with two sets of axes: the original x and y, and the rotated X and Y.

Key Concepts

Eliminating the xy-term

When dealing with equations that include an \(xy\) term, it's often necessary to perform a rotation of the axes to simplify the equation. Removing the \(xy\) term is essential because it complicates identification of the conic section represented by the equation, such as a hyperbola. The rotation of axes allows us to set the coefficient of the \(XY\) term to zero, thus eliminating it.

Here’s how it works:

• The original variables \(x\) and \(y\) are expressed in terms of new variables \(X\) and \(Y\).
• With trigonometric functions of a rotation angle \(θ\), we use the equations \(x = X \cos(θ) - Y \sin(θ)\) and \(y = X \sin(θ) + Y \cos(θ)\).

Substituting these expressions back into the original equation lets us solve for a specific angle \(θ\) that will eliminate the \(XY\) term. This is achieved by solving the equation \(\cos^2(θ) - \sin^2(θ) = 0\), which in our example yields an angle of \(θ = π/4\).

Understanding the Hyperbola

A hyperbola is a type of conic section that is defined by a significant extent of the algebraic equation containing its terms. Once we've rotated the axes, the equation simplifies and reveals the hyperbola's properties.

Here's what you need to know:

• In the canonical form, the hyperbola equation two distinct branches of curves.
• The hyperbola has two asymptotes, which in the case of \(X - Y = 0\), lie directly on the rotated axes.

This means that the hyperbola in our transformed coordinate system becomes aligned with the axes, simplifying graphing and analysis significantly.

Determining the Angle of Rotation

To determine the angle of rotation \(θ\), we rely on trigonometric principles. This angle ensures the \(xy\) term is removed from the equation.

The main equation used is \(\cos^2(θ) - \sin^2(θ) = 0\), derived from the condition that the coefficient of \(XY\) must be zero. Solving this equation confirms that:

• \(θ = π/4\) or \(θ = 3π/4\), both of which are valid for many cases in mathematics.

For convenience, we select \(θ = π/4\). This angle adjusts the system to simplify the equation without the \(XY\) term, making it clearer and more manageable for subsequent plotting.

Trigonometric Equation in Rotation

The trigonometric equation \(\cos^2(θ) - \sin^2(θ) = 0\) plays a pivotal role in the rotation process of our axes.

• It arises from setting the sum and difference of squares equation derivatives of \(XY\) zero.
• The solutions of this equation dictate the specific rotation angles needed to eliminate \(XY\).

For our specific example, both \(π/4\) and \(3π/4\) are solutions, though we choose \(π/4\) for simplicity. This adjustment reveals the pure form of the simplified conic section. Trigonometric identities are fundamental in evaluating these solutions seamlessly to ensure effective axis transformation. Through these identities, the elimination of the \(xy\) term becomes straightforward, allowing a clean transition to a system where analyzing and sketching the curve becomes straightforward.