Problem 13 - 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. $$x y=8$$

Step-by-Step Solution

Answer

The graph of \(xy = 8\) is a hyperbola.

Step 1: Identify the Form of the Equation

The given equation is a second-degree equation: \(xy = 8\). This is an equation of the form \(A = 0\), \(B = 1\), \(C = 0\), \(D = 0\), \(E = 0\), and \(F = -8\). Here, \(A, B, C\) are the coefficients of \(x^2, xy, y^2\), respectively, and \(D, E, F\) are the linear terms and constant term.

Step 2: Apply the Discriminant to Classify the Conic

Calculate the discriminant \(B^2 - 4AC\). For \(xy = 8\), \(B = 1\), \(A = 0\), \(C = 0\), so the discriminant is \(1^2 - 4 \times 0 \times 0 = 1\). Because \(B^2 - 4AC = 1 > 0\), the equation represents a hyperbola.

Step 3: Rotate Axes to Eliminate the \(xy\)-term

To eliminate the \(xy\)-term, we use the rotation formulas: \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\). Choose \(\theta\) such that \(\tan 2\theta = \frac{B}{A-C}\), which is \(\tan 2\theta = \frac{1}{0}\) here, suggesting \(\theta = 45^\circ\). As such, \(x = \frac{x' - y'}{\sqrt{2}}\) and \(y = \frac{x' + y'}{\sqrt{2}}\). After substituting back the new variables, the equation becomes \(\frac{x'^2 - y'^2}{2} = 8\), or simplified, \(x'^2 - y'^2 = 16\).

Step 4: Sketch the Graph

The simplified equation \(x'^2 - y'^2 = 16\) represents a hyperbola centered at the origin with axes aligned with the rotated \(x'\) and \(y'\) axes. Sketched on the new axes, this is a standard hyperbola opening along the \(x'\)-axis.

Key Concepts

Discriminant

The discriminant, often denoted as \(B^2 - 4AC\), is a crucial tool in classifying conic sections. When analyzing a conic equation such as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant helps identify whether the graph represents an ellipse, hyperbola, or parabola. If \(B^2 - 4AC > 0\), the equation is a hyperbola; if \(B^2 - 4AC = 0\), it indicates a parabola; and if \(B^2 - 4AC < 0\), the equation is that of an ellipse.
In the given problem, since the discriminant calculation yields 1, which is greater than 0, the graph is that of a hyperbola. This simple test can save steps, helping students quickly determine the nature of a conic section.

Hyperbola

A hyperbola is one of the distinct shapes in conic sections, characterized by two open curves. It occurs when the plane cuts both double cones in such a way that it doesn't pass through the apex.

Key features of a hyperbola include:

• Two separate branches or curves.
• Asymptotes that the branches approach but never touch.
• A central point known as the center of the hyperbola.

In the context of the equation \(xy = 8\), once simplified and rotated, it results in a hyperbola where the branches open along the new orientation of axes. Understanding these characteristics helps when considering rotation or translating hyperbolas.

Rotation of Axes

The rotation of axes is a technique used to simplify conic sections, especially when eliminating the \(xy\)-term, which complicates the graph. By rotating the coordinate system, the equation can be transformed into a simpler one.
For example, the formulas:\[ x = x'\cos\theta - y'\sin\theta \] \[ y = x'\sin\theta + y'\cos\theta \] are used to switch to a rotated coordinate system. The angle \(\theta\) is chosen such that \(\tan 2\theta = \frac{B}{A-C}\). Upon applying these changes, the original conic can be expressed in its simpler form, making it easier to graph.
In the given exercise, choosing \(\theta = 45^\circ\) successfully eliminates the \(xy\)-term, transforming the equation into a standard hyperbola.

Graphing Conics

Graphing conics is simplified when the equation is in a recognizable form. Conics include circles, ellipses, parabolas, and hyperbolas. After determining the conic type using the discriminant, and possibly rotating the axes, you proceed to sketch.
The graph of a hyperbola, like \(x'^2 - y'^2 = 16\), is centered at the origin in rotated axes and exhibits a familiar form with principal and transverse axes. Each conic type has unique properties that aid in accurate rendering of the graph, such as:

• Vertices and foci help position ellipses and hyperbolas.
• Directrix and axis of symmetry guide parabolas.

Mastering these concepts can enable students to visualize mathematical problems more effectively, ultimately building a better intuitive understanding of conic sections.