Problem 14 - Algebra and Trigonometry

Question

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x y+4=0$$

Step-by-Step Solution

Answer

The graph is a hyperbola rotated by 45 degrees.

Step 1: Identify the general form

The given equation is \( xy + 4 = 0 \). This equation does not directly match the general second-degree equation form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Re-write it as \( 0x^2 + 1xy + 0y^2 + 0x + 0y + 4 = 0 \), which gives the coefficients: \( A = 0 \), \( B = 1 \), \( C = 0 \).

Step 2: Use the discriminant to classify the conic

The discriminant is given by the formula \( B^2 - 4AC \). For this equation, \( B = 1 \), \( A = 0 \), and \( C = 0 \). Compute the discriminant: \( (1)^2 - 4(0)(0) = 1 \). Since the discriminant \( B^2 - 4AC = 1 \) is greater than zero, the graph represents a hyperbola.

Step 3: Use rotation of axes to eliminate the \( xy \)-term

For \( B e 0 \), a rotation of axes is needed to eliminate the \( xy \)-term. The angle of rotation \( \theta \) is found using \( \tan 2\theta = B / (A - C) = 1/0 \), which means \( \theta = \pi/4 \) or \( 45° \). Under this rotation, use new variables \( x' \) and \( y' \): \[ x = x' \cos(\pi/4) - y' \sin(\pi/4) \]\[ y = x' \sin(\pi/4) + y' \cos(\pi/4) \]Replacing \( x \) and \( y \) in the original equation gives \( x'^2 - y'^2 = -8 \).

Step 4: Simplify and sketch the hyperbola

The equation \( x'^2 - y'^2 = -8 \) can be rewritten as \( y'^2 - x'^2 = 8 \), representing a vertically-opening hyperbola. The center of the hyperbola is at the origin \((0,0)\) in the rotated coordinates \( (x', y') \). Draw the hyperbola opening upwards and downwards from this point.

Key Concepts

Discriminant

In conic sections, the discriminant helps us determine the nature of the graph represented by a second-degree equation. The discriminant is defined as \( B^2 - 4AC \). It derives from the general form of a second-degree equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here's how it classifies the conic shapes:

• If the discriminant \( B^2 - 4AC \) is greater than zero, the conic is a hyperbola.
• If it equals zero, the conic is a parabola.
• If it is less than zero, the formation is an ellipse (which includes circles).

In this particular problem, the coefficients \( A = 0 \), \( B = 1 \), and \( C = 0 \), compute to a discriminant of 1, which is greater than zero. This confirms that the graph is a hyperbola.

Rotation of Axes

The rotation of axes is an essential tool to simplify the equations of conic sections by eliminating the \( xy \)-term. This makes graphing and analyzing conics easier and more intuitive. When an equation has a non-zero \( xy \)-term, it can be cumbersome to work with.To eliminate it, we perform a rotation by an angle \( \theta \) defined by the relation \( \tan 2\theta = \frac{B}{A - C} \). The goal is to rewrite the conic equation without the \( xy \)-component using a new set of variables \( x' \) and \( y' \):

• \( x = x' \cos \theta - y' \sin \theta \)
• \( y = x' \sin \theta + y' \cos \theta \)

For the exercise at hand, \( \theta = \pi/4 \) (or \( 45° \)) results in the elimination of the \( xy \)-term, transforming the equation into a form suitable for further analysis.

Hyperbola

A hyperbola is one of the prominent conic shapes formed when a plane intersects both nappes of a double cone. It consists of two disconnected, symmetrical pieces called branches, which open either horizontally or vertically.In a standard form, a hyperbola looks like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for a horizontally-opening hyperbola or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) for a vertical one. The hyperbola's center is at the intersection of its axes of symmetry.From the problem, after appropriate transformations, the equation \( y'^2 - x'^2 = 8 \) indicates a vertically-opening hyperbola. Understanding this characteristic aided in accurately graphing the equation's curve, where the branches extended upwards and downwards from the center at the origin \((0,0)\) in the rotated coordinates.

Second-Degree Equation

Second-degree equations, also known as quadratic or polynomial equations of degree two, play a large role in conic sections. These equations take the general form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The placement and value of these coefficients \( A, B, C \), among others, directly affect the type of conic section the equation describes (such as parabolas, ellipses, or hyperbolas). When solving these equations, the main goals are to identify the type of conic section and simplify the components for easier plotting and analysis. It might involve finding the discriminant or performing a rotation of axes. In this exercise, the original equation \( xy + 4 = 0 \) was recast into this form with zero coefficients for \( x^2 \), \( y^2 \), \( x \), and \( y \). These steps helped unveil its true identity as a hyperbola through subsequent mathematical analyses such as using the discriminant.