Problem 17 - Algebra and Trigonometry

Question

\(15-28=(a)\) 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. $$ x^{2}+2 \sqrt{3} x y-y^{2}+2=0 $$

Step-by-Step Solution

Answer

(a) The conic is a hyperbola. (b) Rotate axes by \( \frac{\pi}{6} \). (c) Sketch the hyperbola.

Step 1: Identify the Type of Conic Section Using the Discriminant

The general form of a conic section is given by \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. \]For the equation given: \[ x^2 + 2\sqrt{3}xy - y^2 + 2 = 0, \]the coefficients are: \( A = 1 \), \( B = 2\sqrt{3} \), \( C = -1 \), \( D = 0 \), \( E = 0 \), and \( F = 2 \).The discriminant \( \Delta \) is calculated as:\[ \Delta = B^2 - 4AC. \]Substitute the values into the formula:\[ \Delta = (2\sqrt{3})^2 - 4 \cdot 1 \cdot (-1) = 12 + 4 = 16. \]Since \( \Delta > 0 \), the graph is a hyperbola.

Step 2: Rotate the Axes to Eliminate the \( xy \)-Term

To eliminate the \( xy \)-term, we apply a rotation of axes. The formula for the angle \( \theta \) such that \( \tan(2\theta) = \frac{B}{A-C} \).Substitute the given values:\[ \tan(2\theta) = \frac{2\sqrt{3}}{1 - (-1)} = \frac{2\sqrt{3}}{2} = \sqrt{3}. \]This gives \( 2\theta = \frac{\pi}{3} \), so \( \theta = \frac{\pi}{6} \).The rotation formulas are:\[ x = x' \cos\theta - y' \sin\theta, \]\[ y = x' \sin\theta + y' \cos\theta. \]Plug \( \theta = \frac{\pi}{6} \):\[ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin\frac{\pi}{6} = \frac{1}{2}. \]Substitute these into the rotation equations:\[ x = x' \frac{\sqrt{3}}{2} - y' \frac{1}{2}, \]\[ y = x' \frac{1}{2} + y' \frac{\sqrt{3}}{2}. \]These equations transform the original conic to one without the \( xy \)-term.

Step 3: Simplify and Graph the Hyperbola

After performing the rotation, the new equation representing the hyperbola does not have an \( xy \)-term. The equation simplifies to a standard form of a hyperbola that can be sketched based on the transformed coordinates.The exact form depends on further simplification, but generally, it is sufficient to plot key points, such as vertices and asymptotes, based on identifying the center and axes via coefficients or calculation.

Key Concepts

Discriminant

In mathematics, especially algebra, the discriminant helps us determine the type of conic section represented by a given equation. For the general conic equation, \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \]there's a special formula for the discriminant, given by:\[ \Delta = B^2 - 4AC. \]

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse.

In the exercise equation \( x^2 + 2\sqrt{3}xy - y^2 + 2 = 0 \), the discriminant \( \Delta \) calculates to 16, indicating it is a hyperbola, where \( \Delta > 0 \). This simple check with the discriminant offers a quick way to classify and analyze conic sections without complex graphing.

Rotation of Axes

The rotation of axes is a technique applied to simplify conic equations, especially useful in eliminating the \( xy \)-term, which can complicate graphing and analysis. The process involves rotating the coordinate system by an angle \( \theta \). For our equation, to remove the \( xy \)-term, we utilize:\[ \tan(2\theta) = \frac{B}{A-C}. \]Plugging in the parameters, \[ \tan(2\theta) = \frac{2\sqrt{3}}{2} = \sqrt{3}, \] relates to an angle where \( 2\theta = \frac{\pi}{3} \), so \( \theta = \frac{\pi}{6} \).Next, the new equations for \( x \) and \( y \) in terms of \( x' \) and \( y' \) are:

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

Using \( \cos\theta = \frac{\sqrt{3}}{2} \) and \( \sin\theta = \frac{1}{2} \), substituting these values translates the system, removing the \( xy \)-term. This makes it more intuitive to analyze the equation's symmetry and graph.

Hyperbola

A hyperbola is one of the conic sections and is defined by its divergent twin branches. It occurs naturally when \( \Delta > 0 \) in the conic's discriminant formula. The standard form of a hyperbola centers around its equations' real and imaginary axes.Key characteristics of a hyperbola include:

• Two separate curves called branches.
• Asymptotes that the branches approach but never meet.
• A center point, often the origin in simplified form.
• Vertices located on each branch, showcasing the closest interaction with the asymptotes.

When graphing a hyperbola, identifying these parts is crucial. It helps illustrate the shape accurately, wherein each branch opens around the shared center and along the x or y-axis, depending on the hyperbola's alignment in the equation. Understanding these features can clarify many of the hyperbola's unique properties.

Conic Section Graphing

Graphing conic sections like parabolas, ellipses, and hyperbolas demands an understanding of their algebraic forms and specific characteristics. Each conic type carries distinct features influencing its sketch.
Hyperbolas in particular require plotting key elements:

• Vertices: The points where each branch is nearest to the center of the hyperbola.
• Center: Located equidistant between vertices.
• Asymptotes: Lines that guide the curve's branching paths.

By using the transformed conic equation post-rotation, we remove complex terms, making these features easier to locate.
For the given equation, translating after rotation aligns the hyperbolic graph symmetrically about its center. From there, sketching becomes more straightforward, and any subsequent graphical analysis, like intersections or distances, is simplified.