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

Step-by-Step Solution

Answer

The graph is a hyperbola. After rotating the axes, sketch the hyperbola centered at the origin.

Step 1: Identify coefficients

For the given equation \( \sqrt{3}x^2 + 3xy = 3 \), identify the coefficients: \( A = \sqrt{3} \), \( B = 3 \), \( C = 0 \). These will be used to determine the type of conic section.

Step 2: Calculate the Discriminant

The discriminant of a conic section is given by: \( B^2 - 4AC \). Substitute the coefficients into the formula: \( Discriminant = 3^2 - 4 \times \sqrt{3} \times 0 = 9 \).

Step 3: Determine the Type of Conic

Since the discriminant \( B^2 - 4AC = 9 > 0 \), the conic section is a hyperbola.

Step 4: Rotation of Axes Formula

To eliminate the \(xy\)-term, use the rotation of axes formula: \( \tan(2\theta) = \frac{B}{A-C} \). Plug the values into the formula: \( \tan(2\theta) = \frac{3}{\sqrt{3} - 0} = \sqrt{3} \).

Step 5: Find the Rotation Angle

Since \( \tan(2\theta) = \sqrt{3} \), \( 2\theta = 60^\circ \) (since \( \tan(60^\circ) = \sqrt{3} \)). So, \( \theta = 30^\circ \).

Step 6: Transform the Equation

Apply the rotation of axes: \( x = x' \cos(30^\circ) - y' \sin(30^\circ) \) and \( y = x' \sin(30^\circ) + y' \cos(30^\circ) \). Substitute into the original equation and simplify to get: \( \frac{3}{2} (x')^2 - (y')^2 = 3 \).

Step 7: Sketch the Graph

The transformed equation \( \frac{3}{2} (x')^2 - (y')^2 = 3 \) represents a hyperbola. Center the graph at the origin and sketch the hyperbola, noting the axes' orientation which has been changed by the rotation.

Key Concepts

Discriminant in Conic Sections

The discriminant in conic sections is a handy tool that helps identify the type of conic section represented by a quadratic equation. For any quadratic equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is given by \( B^2 - 4AC \). This value plays a crucial role in determining whether you are dealing with a parabola, ellipse, or hyperbola.

Here's how it works:

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

In our original exercise, the calculation gave us a discriminant of 9, indicating a hyperbola. This distinction is crucial when moving forward with further transformations or graphing strategies.

Rotation of Axes

Rotation of axes is a method used to eliminate the \(xy\)-term from a conic section's equation. It simplifies the graph and equation, making analysis easier. This technique is especially useful when dealing with equations where the conic section's orientation isn't along the traditional x or y axes.

Here's a step-by-step guide:

• Start with the rotation formula: \( \tan(2\theta) = \frac{B}{A-C} \).
• Substitute the coefficients from your equation into the formula. In our example, it gives \( \tan(2\theta) = \sqrt{3} \).
• Find \( \theta \), which is half of the angle given by \( 2\theta \). In this case, \( 2\theta = 60^\circ \), so \( \theta = 30^\circ \).

Apply the transformation to the original equation using the rotated coordinates \( x' \) and \( y' \), which are derived from trigonometric identities. This process allows us to simplify the equation and make sketching or further analysis more intuitive.

Understanding Hyperbolas

Hyperbolas are fascinating conic sections characterized by two separate curves, called branches. Recognizing and understanding their properties are fundamental aspects of conic section studies.

Key characteristics include:

• They are defined as the set of all points where the difference of distances to two fixed points (foci) is constant.
• Hyperbolas can be horizontal or vertical, depending on their orientation.
• They have asymptotes, which are lines the branches approach but never touch.

The equation we got after rotating the axes, \( \frac{3}{2} (x')^2 - (y')^2 = 3 \), is typical for a hyperbola, which further validates the discriminant's indication. Understanding these properties aids in correctly sketching the graph and predicting the axis orientation.

Sketching Graphs

Sketching graphs of conics, like hyperbolas, requires careful plotting and understanding of their general equation and properties. From the transformed equation, \( \frac{3}{2} (x')^2 - (y')^2 = 3 \), you can gather insights on orientation and scale.

Here are the steps to sketch a hyperbola:

• Identify the center, plotted at the origin if the equation is in standard form after rotation.
• Determine the vertices and foci parameters using the derived equation.
• Draw asymptotes based on coefficients, guiding the shape of the branches.
• Sketch the branches, ensuring the distance between them matches the constant difference in the hyperbola definition.

With these steps, even complex hyperbolas can be effectively visualized, enhancing comprehension and application in various mathematical contexts.