Problem 27 - 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. $$52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75$$

Step-by-Step Solution

Answer

The equation represents an ellipse. The rotation of axes transforms it to eliminate the xy-term for sketching.

Step 1: Understanding the General Quadratic Equation

The given equation is in the form of a general quadratic equation: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Here, \( A = 52 \), \( B = 72 \), \( C = 73 \), \( D = -40 \), \( E = 30 \), and \( F = 75 \).

Step 2: Calculation of the Discriminant

The discriminant \( \Delta \) of a conic section is given by the formula: \[ \Delta = B^2 - 4AC \].Substituting our values into the formula: \[ \Delta = 72^2 - 4(52)(73) \].Calculating the values, we find:\[ 72^2 = 5184 \quad \text{and} \quad 4 \times 52 \times 73 = 15184 \].Thus, \[ \Delta = 5184 - 15184 = -10000 \].

Step 3: Interpretation of the Discriminant

For a conic section, if \( \Delta < 0 \), the equation represents an ellipse. Since \( \Delta = -10000 < 0 \), the graph of the given equation is an ellipse.

Step 4: Using Rotation of Axes to Eliminate xy-term

To eliminate the \( xy \)-term, we use the rotation formulas:\[x = X \cos \theta - Y \sin \theta \quad \text{and} \quad y = X \sin \theta + Y \cos \theta\]where \( \tan 2\theta = \frac{B}{A-C} \). Here, \( \tan 2\theta = \frac{72}{52-73} = \frac{72}{-21} \).Solving for \( \theta \), calculate \( \theta \) such that \( \tan \theta = \sqrt{\tan 2\theta} \):\( 2\theta = \tan^{-1}\left(\frac{-72}{21}\right) \). From this, find \( \theta \).

Step 5: Substitute and Simplify the Equation

After calculating \( \theta \), replace \( x \) and \( y \) with new variables \( X \) and \( Y \) using the rotation formulas.Simplify the equation to get it in the standard form of an ellipse without the \( XY \)-term.

Step 6: Sketch the Graph of the Ellipse

Once in standard form, the ellipse can be graphed by identifying its center, axes, and orientation from the simplified equation. Draw the ellipse based on these characteristics.

Key Concepts

Discriminant in Conic Sections

The discriminant helps us classify the type of conic section represented by a quadratic equation. The equation given is in the general form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]To determine the conic section, we compute the discriminant \( \Delta \) as follows:\[ \Delta = B^2 - 4AC \]

• If \( \Delta > 0 \), it's a hyperbola.
• If \( \Delta = 0 \), it's a parabola.
• If \( \Delta < 0 \), it's an ellipse.

In our case, substituting the values \( A = 52 \), \( B = 72 \), and \( C = 73 \) into the formula, we find \( \Delta = 5184 - 15184 = -10000 \).
This negative value indicates that the given equation graph is an ellipse.

Understanding Ellipses

An ellipse is one of the fundamental shapes in the study of conic sections. It is often perceived as an elongated circle, having two main features: a major and a minor axis.

Characteristics of an Ellipse

• Has two focal points (foci), which determine its shape.
• The sum of distances from any point on the ellipse to the two foci is constant.
• The longest axis is called the major axis, and the shortest is the minor axis.

Ellipses arise naturally in various fields, such as planetary orbits in astronomy. Here, after using the discriminant to establish that our equation is an ellipse, we proceed further to simplify the equation by rotating the axes.

Rotation of Axes to Simplify Ellipses

To convert the ellipse equation to its simplest form, we eliminate the \( xy \)-term using a rotation of axes. This is achieved through the transformation formulas:\[ x = X \cos \theta - Y \sin \theta \]\[ y = X \sin \theta + Y \cos \theta \]where the angle \( \theta \) is determined by the equation:\[ \tan 2\theta = \frac{B}{A-C} \]For our specific problem, with \( B = 72 \), \( A = 52 \), and \( C = 73 \), we calculate \( \tan 2\theta = \frac{-72}{21} \).
Solving for \( \theta \), we use trigonometric identities to find the angle, allowing us to rewrite the quadratic equation in terms of new variables \( X \) and \( Y \) without the troublesome \( XY \)-term.
This rotation clarifies the orientation of the ellipse, allowing us to sketch it with its axes aligned along the coordinate axes.