Problem 16 - Precalculus: Mathematical 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 \(x y\) -term. (c) Sketch the graph. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16$$

Step-by-Step Solution

Answer

The equation describes an ellipse.

Step 1: Identify Coefficients

The given equation is \(13x^2 + 6\sqrt{3}xy + 7y^2 = 16\). Identify the coefficients: \(A=13\), \(B=6\sqrt{3}\), \(C=7\).

Step 2: Compute the Discriminant

The discriminant is calculated using the formula \(B^2 - 4AC\). Substitute the values: \((6\sqrt{3})^2 - 4 \cdot 13 \cdot 7\). Calculate \(B^2 = 108\) and \(4AC = 364\). Thus, the discriminant is \(108 - 364 = -256\).

Step 3: Determine the Type of Conic

Based on the discriminant: If \(B^2 - 4AC > 0\), it's a hyperbola. If \(B^2 - 4AC = 0\), it's a parabola. If \(B^2 - 4AC < 0\), it's an ellipse. Since \(B^2 - 4AC = -256 < 0\), the equation represents an ellipse.

Step 4: Find the Rotation Angle

To eliminate the \(xy\)-term, use the formula \(\tan(2\theta) = \frac{B}{A-C}\). Substitute the values: \(\tan(2\theta) = \frac{6\sqrt{3}}{13-7}\). Thus, \(\tan(2\theta) = \frac{6\sqrt{3}}{6} = \sqrt{3}\) implying \(2\theta = 60\degr\) or \(\theta = 30\degr\).

Step 5: Apply Rotation

The transformation equations with rotation are \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\). Substitute \(\theta = 30\degr\): \(x = x'\frac{\sqrt{3}}{2} - y'\frac{1}{2}\), \(y = x'\frac{1}{2} + y'\frac{\sqrt{3}}{2}\). Substitute these into the original equation to eliminate the \(xy\)-term.

Step 6: Simplify the New Equation

After substitution, you'll reach a new equation in \(x'^2\) and \(y'^2\) without the \(x'y'\) term. Simplify it considering any constant adjustments from transformation.

Step 7: Graph the Resulting Equation

Sketch the graph of the ellipse using the simplified form determined in Step 6. Identify axes based on transformation, maintaining the ellipse's orientation.

Key Concepts

Understanding the Discriminant

In the context of conic sections, the discriminant tells us about the nature of the curve defined by a quadratic equation. For a general equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is given by the formula \(B^2 - 4AC\). The value it takes helps classify the conic:

• If \(B^2 - 4AC > 0\), the graph represents a hyperbola.
• If \(B^2 - 4AC = 0\), the graph is that of a parabola.
• If \(B^2 - 4AC < 0\), the graph is an ellipse.

Given the equation from the exercise, we calculated the discriminant to be \(-256\), which is less than zero. Hence, the equation represents an ellipse.The discriminant helps us quickly determine the type of conic section without graphing, saving time and effort. Always remember to check the sign of the result carefully because even a small error could lead to a different type of curve.

Exploring Ellipses

An ellipse is one of the basic types of conic sections and is shaped like an elongated circle. In simple terms, it is defined as the set of all points for which the sum of the distances from two fixed points (foci) is constant. Ellipses have several interesting properties:

• The longest diameter across the ellipse is the major axis, while the shortest is the minor axis.
• The points where the axes intersect are the center of the ellipse.

One common equation form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h,k)\) is the center and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. Understanding the properties of an ellipse, such as its axes and foci, can be crucial for applications in areas ranging from physics to engineering. In the exercise, recognizing the equation describes an ellipse allows us to apply appropriate transformations, like the rotation of axes, to better understand or simplify the equation.

Rotation of Axes

Rotation of axes is a technique used to simplify conic sections by removing the \(xy\)-term. For an equation like \(Ax^2 + Bxy + Cy^2 = D\), the presence of the \(xy\)-term complicates graphing because it implies that the conic is rotated. To "un-rotate" the conic, we apply a rotation of axes. This involves:

• Finding the angle \(\theta\) using \(\tan(2\theta) = \frac{B}{A-C}\).
• Using the trigonometric equations to transform coordinates. For example:
  • \(x = x'\cos\theta - y'\sin\theta\)
  • \(y = x'\sin\theta + y'\cos\theta\)

In the exercise, this process involved finding \(\theta = 30\degree\) and substituting these transformations to eliminate the \(xy\)-term.After performing this transformation, the equation typically simplifies to one without the \(xy\)-term, making it easier to sketch or analyze. This method is powerful because it can convert complex conic sections into a more standard form, facilitating better understanding and interpretation.