Problem 23 - 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. $$2 \sqrt{3} x^{2}-6 x y+\sqrt{3} x+3 y=0$$

Step-by-Step Solution

Answer

The equation represents a hyperbola, as determined by the positive discriminant.

Step 1: Identify Equation Coefficients

The given equation is \( 2 \sqrt{3} x^{2} - 6 x y + \sqrt{3} x + 3 y = 0 \). The general form of a conic equation is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here, \( A = 2 \sqrt{3} \), \( B = -6 \), \( C = 0 \).

Step 2: Calculate Discriminant

The discriminant for a conic is given by \( B^2 - 4AC \). Substitute the values to get \( (-6)^2 - 4(2\sqrt{3})(0) = 36 - 0 = 36 \).

Step 3: Determine Type of Conic

Based on the discriminant: if \( B^2 - 4AC > 0 \) it represents a hyperbola, if \( B^2 - 4AC = 0 \) a parabola, and if \( B^2 - 4AC < 0 \) an ellipse. Here, \( 36 > 0 \), hence it is a hyperbola.

Step 4: Find Rotation Angle to Eliminate \(xy\)-term

Use \( \tan(2\theta) = \frac{B}{A-C} \). Substituting \( B = -6 \), \( A = 2\sqrt{3} \), and \( C = 0 \), we have \( \tan(2\theta) = \frac{-6}{2\sqrt{3}} = -\sqrt{3} \). This implies \( 2\theta = 240^\circ \) or \( 300^\circ \), so \( \theta = 120^\circ \) or \( 150^\circ \).

Step 5: Transform Variables

Substituting \( x = X \cos \theta - Y \sin \theta \) and \( y = X \sin \theta + Y \cos \theta \), apply \( \theta = 120^\circ \) or \( 150^\circ \) to eliminate the \( xy \) term in the equations. Rewriting and solving the transformed equation will reflect axes without the \( xy \)-term.

Step 6: Simplify and Sketch

After transforming, simplify the equation to standard hyperbolic form, such as \( \frac{X^2}{a^2} - \frac{Y^2}{b^2} = 1 \), and sketch the graph using the standard appearance of a hyperbola, indicating the axes swap and rotations taken place.

Step 7: Conclude with Graph Characteristics

Confirm graph is a hyperbola using derived formula's shape. Ensure axes reflect correct orientation to visualize data from transformations and present a clear sketch, showing hyperbola's curves.

Key Concepts

Discriminant

The discriminant is a mathematical expression that helps us determine the nature of conic sections based on their equation coefficients. For a general second-degree equation in the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is given by the formula \( B^2 - 4AC \).

• When the discriminant \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), it indicates a parabola.
• Finally, if \( B^2 - 4AC < 0 \), the conic is an ellipse.

In our exercise, the given equation was converted into standard form coefficients: \( A = 2\sqrt{3} \), \( B = -6 \), and \( C = 0 \). By substituting these into the discriminant formula, we calculated \( (-6)^2 - 4(2\sqrt{3})(0) = 36 - 0 = 36 \). The result was greater than zero, confirming the graph is a hyperbola.

Rotation of Axes

Sometimes a conic section's equation involves an \( xy \)-term, which complicates the graphing process. By using a rotation of axes, you can eliminate the \( xy \)-term to simplify the equation. This helps in identifying and sketching the conic section more easily.
To find the angle of rotation, use the formula \( \tan(2\theta) = \frac{B}{A-C} \). In our exercise, substituting \( B = -6 \), \( A = 2\sqrt{3} \), and \( C = 0 \), we get \( \tan(2\theta) = -\sqrt{3} \). Solving for \( \theta \) gives us possible angles of \( 120^\circ \) or \( 150^\circ \).
These angles can be used to transform the coordinate system. By expressing \( x \) and \( y \) in terms of new coordinates, \( X \) and \( Y \), through transformation formulas, the new equation that emerges will generally lack the \( xy \)-term, making the graph easier to handle.

Hyperbola

A hyperbola is a type of conic section that is defined by its distinct pair of curves, known as branches, that open in opposite directions. Its standard form, after transformation, usually appears as \( \frac{X^2}{a^2} - \frac{Y^2}{b^2} = 1 \). This equation highlights the attributes of the hyperbola:

• Each branch extends infinitely in opposite directions.
• The asymptotes, which guide the shape, intersect at the center of the hyperbola and bisect the angle between the two branches.
• They have two focal points, and the distance between any point on the hyperbola to the two foci is constant.

In the context of our exercise, after using rotation of axes and eliminating the \( xy \)-term, the transformed equation's structure allows us to identify these key features. This transformation not only simplifies graphing but clarifies the hyperbola's behavior.

Elimination of xy-term

The \( xy \)-term in a conic section's equation signals rotation in the plane, complicating direct graph analysis. To eliminate this term, we rotate the coordinate system by a specific angle. This adjustment streamlines the conic's equation to help us visualize and graph accurately.The process involves:

• Computing \( \tan(2\theta) \) by \( \frac{B}{A-C} \), which gives us the rotation angle \( \theta \).
• Applying transformations such as \( x = X \cos \theta - Y \sin \theta \) and \( y = X \sin \theta + Y \cos \theta \).
• Rewriting the equation in these new terms ensures the \( xy \)-term is removed.

In our exercise, choosing \( \theta \) from the calculated angles and applying the transformations allows us to restate the hyperbola equation minus the \( xy \)-term. This step greatly aids in clarifying the geometry and orientation of the graph.