Problem 14 - Precalculus with Limits

Question

In Exercises 13-26, rotate the axes to eliminate the \(xy\)-term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. \(xy - 4 = 0\)

Step-by-Step Solution

Answer

The rotation of the axes doesn't change the equation in this case, so the result is the same equation, \( xy - 4 = 0 \). The graph is a set of hyperbola segments in the first and third quadrants, which are mirrored to each other through the origin.

Step 1: Compute the Angle of Rotation

The angle of rotation, \( \theta \), can be calculated using the formula \( \theta = \frac{1}{2} \arctan \left( \frac{b}{a - c} \right) = \frac{1}{2} \arctan \left( \frac{-1}{0} \right) = 0 \) where \( a = c = 0 \) because there are no \( x^{2} \) or \( y^{2} \) terms, and \( b = -1 \) from the term \( -xy \).

Step 2: Replace x and y with rotated axes

Secondly, We will replace \( x \) and \( y \) in the equation with the rotated axes using the formulas \( x = X \cos(\theta) - Y \sin(\theta) \) and \( y = X \sin(\theta) + Y \cos(\theta) \). After substituting \( X \) and \( Y \) for \( x \) and \( y \) we will get \( xy - 4 = 0 \) as \( X \cdot Y - 4 = 0 \) again as \( \theta = 0 \).

Step 3: Rewrite in Standard Form

As the equation \( X \cdot Y - 4 = 0 \) is already the simplest standard form of the hyperbola, no further simplification is required.

Step 4: Draw the graph

The graph is a hyperbola where the asymptotes are the coordinate axes. The values of X and Y are distributed on two lines that mirror each other across both the X and Y axes.

Key Concepts

Hyperbola

A hyperbola is a unique shape that arises from slicing a double-cone with a plane in specific ways. Unlike circles, ellipses, or parabolas, hyperbolas have two separate curved lines or branches. This results in a very distinct appearance. Hyperbolas are defined by an expression of the form

• \( rac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

Note how it includes a subtraction, unlike the addition found in circles or ellipses.
Hyperbolas have asymptotes, which are lines that the hyperbola approaches but never touches. These asymptotes give the hyperbola its characteristic open shape.
In the given exercise, the equation generated after coordinate rotation reveals a hyperbola. This is visible in the equation:

• \( X\cdot Y - 4 = 0 \)

This specific form is a simple representation of a hyperbola.

Standard Form

Standard form is an essential concept in mathematics. It provides a uniform way to express equations.
This makes it easier not just to understand but also to graph and analyze them. For conic sections like hyperbolas, the standard form is crucial.
The standard form for a hyperbola that has its center at the origin can be:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \)

These forms cover horizontal and vertical hyperbolas, respectively.
The standard form tells us several properties about the hyperbola, like the direction opening, and helps in determining the asymptotes.
In the exercise, the equation \( X \cdot Y - 4 = 0 \) acts as a hyperbola's standard form, showing a hyperbola with its diagonals as asymptotes.

Angle of Rotation

The angle of rotation is a critical step when modifying coordinate systems, especially when trying to eliminate the \(xy\)-term.
This procedure in equations like the one given simplifies problem-solving by aligning the equations to the new axes.
To calculate the angle of rotation \(\theta\), use:

• \( \theta = \frac{1}{2} \arctan \left( \frac{b}{a-c} \right) \)

Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
If the \(xy\)-term exists, the equation represents a tilt, which is corrected by the right rotation angle.
In the exercise, we set \(\theta\) to zero due to the structure of the equation, \(-xy\). This angle choice shows that no rotation is necessary to simplify the given equation into standard form.