Problem 18 - 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. $$21 x^{2}+10 \sqrt{3} x y+31 y^{2}=144$$

Step-by-Step Solution

Answer

The equation describes an ellipse. After rotation, it is \(u^2 + 3v^2 = 96\). Sketch the ellipse with semi-axes \(4\sqrt{6}\) and \(4\sqrt{2}\).

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

The general form of a conic section is given by the equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The discriminant \(D\) is calculated as \(B^2 - 4AC\). For the given equation, we have \(A=21\), \(B=10\sqrt{3}\), and \(C=31\). Calculate the discriminant: \(D = (10\sqrt{3})^2 - 4 \times 21 \times 31\). Simplifying, \(D = 300 - 2604 = -2304\). Since \(D < 0\), the graph is an ellipse.

Step 2: Eliminate the XY-term using Rotation of Axes

To eliminate the \(xy\)-term, perform a rotation of axes. The angle of rotation \(\theta\) is given by \(\tan(2\theta) = \frac{B}{A-C}\). Here, \(\tan(2\theta) = \frac{10\sqrt{3}}{21-31} = \frac{10\sqrt{3}}{-10} = -\sqrt{3}\). Thus, \(2\theta = 120^\circ\), so \(\theta = 60^\circ\). Use the rotation formulas to transform the equation, substituting for \(x\) and \(y\) using \(x = x' \cos(\theta) - y' \sin(\theta)\) and \(y = x' \sin(\theta) + y' \cos(\theta)\). This yields the transformed equation \(u^2 + 3v^2 = 96\), where \(u\) and \(v\) are the rotated coordinates.

Step 3: Sketch the Graph

The transformed equation \(u^2 + 3v^2 = 96\) represents an ellipse. Rewrite it as \(\frac{u^2}{96} + \frac{v^2}{32} = 1\), which is in standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The semi-major axis length is \(\sqrt{96} = 4\sqrt{6}\) and the semi-minor axis length is \(\sqrt{32} = 4\sqrt{2}\). Sketch an ellipse centered at the origin, elongated along the \(v\)-axis.

Key Concepts

Ellipse

An ellipse is a type of conic section that appears as an elongated circle, often described as an oval. In mathematical terms, an ellipse is the set of all points such that the sum of the distances from two fixed points (called foci) is constant. The equation of an ellipse in standard form is given by:

• Horizontal: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Vertical: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

The parameters \(h\) and \(k\) represent the coordinates of the center of the ellipse, while \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively. The semi-major axis is the longest diameter that runs through the center, and the semi-minor axis runs perpendicular to the semi-major along the shorter diameter. When we deal with problems involving ellipses, it's crucial to recognize these principal dimensions, as they dictate the overall shape and orientation of the ellipse. For example, in the provided solution, the semi-major and semi-minor axes are found from the equation \( \frac{u^2}{96} + \frac{v^2}{32} = 1 \), leading to major and minor axes of \(4\sqrt{6}\) and \(4\sqrt{2}\), respectively.

Discriminant in Conic Sections

The discriminant in conic sections is an essential tool in determining the type of conic section from a general second-degree equation. The general form is given by: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The discriminant \(D\) is calculated using the formula: \( D = B^2 - 4AC \).

This discriminant helps us quickly determine whether we are dealing with an ellipse, parabola, or hyperbola:

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

In the given exercise, we computed the discriminant as \( D = (10\sqrt{3})^2 - 4 \times 21 \times 31 \). This simplifies to \( D = 300 - 2604 = -2304 \). Since \( D < 0 \), this confirms that the conic section is indeed an ellipse. Recognizing the type of conic section helps to set the stage for further transformation and analysis of the graph.

Rotation of Axes

The rotation of axes is a mathematical transformation used to simplify equations by eliminating the \(xy\)-term from conic section equations. Often, the \(xy\)-term complicates the recognition and analysis of the conic section. By rotating the coordinate axes by an appropriate angle, we can transform the equation into a simpler form.

The angle \(\theta\) for the rotation is found using:\[ \tan(2\theta) = \frac{B}{A-C} \]Once \(\theta\) is found, the coordinates are rotated using:

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

In the exercise, the \(xy\)-term is eliminated using a rotation angle \(\theta = 60^\circ\), calculated from \( \tan(2\theta) = -\sqrt{3} \). This transforms the original equation into \(u^2 + 3v^2 = 96\), a recognizable ellipse equation. The rotation of axes is a handy procedure to simplify complex equations and make it easier to plot or analyze the conic sections.