Problem 20 - Precalculus: Mathematics for Calculus

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 xy-term. (c) Sketch the graph. $$21 x^{2}+10 \sqrt{3} x y+31 y^{2}=144$$

Step-by-Step Solution

Answer

(a) The equation represents an ellipse. (b) Rotate axes by 30° to eliminate xy-term. (c) The graph is an ellipse centered at the origin.

Step 1: Identifying the type of conic section using the discriminant

The general form of a conic section is given by the equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, we have \(A = 21\), \(B = 10\sqrt{3}\), \(C = 31\), \(D = 0\), \(E = 0\), \(F = -144\). The discriminant \( \Delta \) is calculated as \( B^2 - 4AC \). Substituting the given values, we get \( \Delta = (10\sqrt{3})^2 - 4(21)(31) = 300 - 2604 = -2304 \). Since \( \Delta < 0 \), the equation represents an ellipse.

Step 2: Rotation of axes to eliminate the xy-term

To eliminate the \(xy\)-term, we use the rotation formulas: \( x = x' \cos \theta - y' \sin \theta \) and \( y = x' \sin \theta + y' \cos \theta \), where \( \tan(2\theta) = \frac{B}{A-C} = \frac{10\sqrt{3}}{21 - 31} = -\sqrt{3} \). Solving, \( 2\theta = 120^\circ \) or \( \theta = 60^\circ \). Using \( \theta = 30^\circ \) (since we need \( \theta = 60^\circ / 2 \)), we have \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \sin \theta = \frac{1}{2} \). Substitute these values into the rotation formulas to convert the equation into a new system where the \(xy\)-term will vanish.

Step 3: Simplifying the rotated equation

After substituting the rotation formulas into the conic equation, we get a new equation with no \(xy\)-term. Simplification involves substituting \( x = x' \cos(30^\circ) - y' \sin(30^\circ) \) and \( y = x' \sin(30^\circ) + y' \cos(30^\circ) \) into the equation and comparing coefficients. The simplified form will look like \( A'x'^2 + C'y'^2 = 144 \), representing the standard form of an ellipse.

Step 4: Sketching the graph

The equation now represents an ellipse, whose axes have been aligned to the coordinate axes due to the rotation. Using the coefficients from the simplified equation (which are determined in the previous step), identify the lengths of the semi-major and semi-minor axes. The center of the ellipse remains at the origin (0, 0) due to the lack of linear terms.

Key Concepts

Discriminant

In the world of conic sections, the discriminant is a powerful tool that helps determine the type of conic represented by a given equation. For any conic section in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is defined as \[\Delta = B^2 - 4AC\].

The value of \(\Delta\) plays a crucial role in identifying the conic section:

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

Given the equation in our exercise, by substituting \(A = 21\), \(B = 10\sqrt{3}\), and \(C = 31\) into the discriminant formula, we calculate \(\Delta = (10\sqrt{3})^2 - 4(21)(31) = -2304\).

Since \(\Delta < 0\), the conic section is an ellipse. Understanding these basic properties of the discriminant allows students to efficiently categorize different conic sections.

Rotation of Axes

The rotation of axes is a mathematical technique used to simplify conic equations by eliminating the \(xy\)-term. This is especially useful in transforming the equation of a conic section to align its axes with the coordinate axes.

The rotation involves changing the coordinate system using the formulas:
\[x = x' \cos \theta - y' \sin \theta\]
\[y = x' \sin \theta + y' \cos \theta\]
where \(\theta\) is the angle of rotation.

To find \(\theta\), use the equation \(\tan(2\theta) = \frac{B}{A-C}\). In our example, this calculates as \(\tan(2\theta) = -\sqrt{3}\), leading to \(2\theta = 120^\circ\) or \(\theta = 60^\circ\). However, to apply the rotation correctly, use \(\theta = 30^\circ\), which is half of \(60^\circ\).

Using \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(30^\circ) = \frac{1}{2}\), substitute these into the rotation formulas to derive a new axis-aligned equation. This process simplifies the original equation and directly leads to understanding the graph's orientation.

Ellipse

An ellipse represents a type of conic section characterized by the equation \(A'x'^2 + C'y'^2 = 144\) after rotation, as shown in the exercise. It's like a stretched circle and has unique properties regarding its geometric shape.

Ellipses have:

• Two axes: a major axis (the longest) and a minor axis (the shortest).
• A center point, which remains at the origin after rotation in our case.
• Focal points along the major axis, inside the boundary of the ellipse.

Once you have the simplified form of the equation, it becomes straightforward to identify the lengths of these axes. The major axis corresponds to the larger coefficient in the standard form.

In this example, the absence of linear terms indicates that the center is precisely at the axes' intersection. Drawing the ellipse involves plotting these axes and ensuring the graph reflects the calculated proportions. Visualizing an ellipse can be easier once it is reduced to its standard form through the rotation process.