Problem 22 - 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. $$153 x^{2}+192 x y+97 y^{2}=225$$

Step-by-Step Solution

Answer

The equation represents an ellipse; use rotation to eliminate the \(xy\) term and sketch an ellipse.

Step 1: Identify the Conic Section

The given equation is \(153x^2 + 192xy + 97y^2 = 225\). To determine the type of conic section, compute the discriminant \(D = B^2 - 4AC\), where \(A = 153\), \(B = 192\), and \(C = 97\). Substitute these values: \(D = 192^2 - 4 \cdot 153 \cdot 97\). Calculate \(D = 36864 - 59244 = -22380\). Since \(D < 0\), the conic section is an ellipse.

Step 2: Find Rotation Angle

To eliminate the \(xy\) term, calculate the angle \( \theta \) using \( \tan(2\theta) = \frac{B}{A-C} \). Substituting the values gives \( \tan(2\theta) = \frac{192}{153 - 97} = \frac{192}{56} = \frac{24}{7}\). Solve for \(2\theta\) using a calculator: \(2\theta = \tan^{-1}(\frac{24}{7})\). Then, find \(\theta = \frac{1}{2}\tan^{-1}(\frac{24}{7})\).

Step 3: Perform Rotation of Axes

Use the rotation formulas: \(x = X\cos\theta - Y\sin\theta\) and \(y = X\sin\theta + Y\cos\theta\). Substitute these into the original equation to eliminate the \(xy\) term. This involves substituting \(x\) and \(y\) into \(153x^2 + 192xy + 97y^2 = 225\) and simplifying to remove the \(xy\) term.

Step 4: Sketch the Graph

After eliminating the \(xy\) term, you should arrive at a simplified equation that represents an ellipse, such as \(AX^2 + CY^2 = D\). Sketch the graph using the ellipse equation, plotting it on the transformed \((X, Y)\) coordinate grid. Consider the orientation change due to rotation.

Key Concepts

Discriminant

The discriminant is a key concept when working with conic sections and helps to identify what type of conic section an equation represents. For a general quadratic equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant \(D\) is calculated as \(D = B^2 - 4AC\). Here’s how it works:

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

In our example, the equation is \(153x^2 + 192xy + 97y^2 = 225\). By substituting into the discriminant formula, we find that \(D = 192^2 - 4 \cdot 153 \cdot 97 = -22380\). Since \(D < 0\), the equation represents an ellipse.

Rotation of Axes

The rotation of axes is a technique used to simplify conic sections that contain an \(xy\)-term. This term can complicate determining the type of conic section and plotting its graph. The objective is to perform a rotation that eliminates the \(xy\)-term, making the equation easier to work with. To rotate the axes, we calculate the angle \(\theta\) using the formula \(\tan(2\theta) = \frac{B}{A-C}\). In our example, we substitute and find \(\tan(2\theta) = \frac{192}{56} = \frac{24}{7}\). Solving \(2\theta = \tan^{-1}(\frac{24}{7})\) and then finding \(\theta = \frac{1}{2} \tan^{-1}(\frac{24}{7})\) gives us the rotation angle. This angle helps transform the original coordinates \((x, y)\) to new coordinates \((X, Y)\).

Ellipse

An ellipse is one of the primary conic sections characterized by its oval shape. In geometry, it is often considered as a set of all points for which the sum of the distances to two fixed points (foci) is constant. After determining the discriminant indicates an ellipse, the next steps often involve rewriting the equation without the \(xy\)-term. Once this simplified equation is obtained, it typically takes the form \(AX^2 + CY^2 = D\), where the terms reflect the axes lengths and orientation. Ellipses occur in nature and practical applications, such as planetary orbits. Recognizing the forms and properties of ellipses helps in numerous scientific fields, providing a deeper understanding of the shapes present in the world.

XY-term

The \(xy\)-term in conic sections can indicate a need for further analysis when trying to determine the type of curve it represents. Its presence typically points towards a rotated conic, mixing the roles of the \(x\) and \(y\) variables compared to simpler circles or parabolas. Removing the \(xy\)-term requires applying rotation of axes, using rotation formulas:

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

By transforming \(x\) and \(y\) to \(X\) and \(Y\), the term gets eliminated, and it's easier to analyze the conic. The elimination simplifies the expressions and helps focus on the primary shape and its properties.