Problem 21 - Algebra and Trigonometry

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

Step-by-Step Solution

Answer

The graph represents a hyperbola.

Step 1: Identify the Equation

Rewrite the given equation \( \sqrt{3}x^2 + 3xy = 3 \) to compare it with the standard conic section equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here, \( A = \sqrt{3}, B = 3, C = 0 \).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) for a conic section is given by \(\Delta = B^2 - 4AC\). Substitute \(A, B,\) and \(C\) into the formula: \(\Delta = 3^2 - 4(\sqrt{3})(0) = 9\).

Step 3: Determine the Type of Conic

The sign of \(\Delta\) determines the type of conic section. Since \(\Delta > 0\), this indicates a hyperbola.

Step 4: Find Rotation Angle

To eliminate the \(xy\)-term, use the formula \(\cot(2\theta) = \frac{A-C}{B}\). With \(A = \sqrt{3}, B = 3, C = 0\), \(\cot(2\theta) = \frac{\sqrt{3} - 0}{3} = \frac{\sqrt{3}}{3}\).

Step 5: Calculate \(\theta\)

Solve \(\cot(2\theta) = \frac{\sqrt{3}}{3}\) for \(2\theta\). Since \(2\theta\) must match a known cotangent value, choose \(2\theta = 30^\circ\), thus \(\theta = 15^\circ\).

Step 6: Transform the Coordinates

Use the rotation formulas: \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\) to change variables. Substitute \(\theta = 15^\circ\).

Step 7: Simplify the New Equation

Substitute the expressions from Step 6 into the original equation \(\sqrt{3}x^2 + 3xy = 3\) and simplify. The XY-term should now vanish, leaving a conic section equation in the \(x'y'\)-plane.

Step 8: Sketch the Graph

Using the transformed equation, sketch the hyperbola in the \(x'y'\)-plane. Identify its key features such as asymptotes and vertex to draw an accurate shape of the hyperbola.

Key Concepts

Understanding the Discriminant in Conic Sections

The discriminant is a valuable tool for identifying the type of conic section an equation represents without graphing it. For a conic section equation in the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]the discriminant \( \Delta \) is calculated using the formula \[ \Delta = B^2 - 4AC \].

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

For the given equation, \( \sqrt{3}x^2 + 3xy = 3 \), with \( A = \sqrt{3} \), \( B = 3 \), and \( C = 0 \), we find \( \Delta = 9 \). Since \( \Delta > 0 \), the graph is a hyperbola. Evaluating the discriminant in this way provides a quick method to classify and understand the nature of the conic section.

Exploring Hyperbolas

A hyperbola is a type of conic section that forms when a plane intersects both nappes of a double cone. Hyperbolas are characterized by two distinct, mirror-image curves called branches.

• Unlike ellipses or parabolas, hyperbolas have a transverse and a conjugate axis.
• The transverse axis passes through the vertices of the hyperbola.
• Asymptotes: Two lines that the curve approaches but never touches. They determine the direction of the branches.

In the Cartesian plane, hyperbolas appear as curved shapes with open ends extending towards infinity. These shapes are symmetric about their center. The primary features of a hyperbola include **vertices** (the closest points on each branch to the center) and **foci**, which lie inside each branch and determine its shape. The equation for a hyperbola can take on various forms, particularly when transforming or rotating the axes.

The Geometry of Rotation of Axes

Sometimes equations of conics include an \(xy\)-term, which can complicate identification and graphing. To simplify, we can use a rotation of axes.

• The goal of this rotation is often to eliminate the \(xy\)-term from the equation.
• For rotation, utilize the formula \( \cot(2\theta) = \frac{A-C}{B} \) to find the rotation angle \( \theta \).

Using our example, where \( A = \sqrt{3} \), \( B = 3 \), and \( C = 0 \), we calculate \( \cot(2\theta) = \frac{\sqrt{3}}{3} \). Solving gives \( \theta = 15^\circ \). With this angle, we rotate using the formulas:- \( x = x'\cos\theta - y'\sin\theta \)- \( y = x'\sin\theta + y'\cos\theta \)This rotation results in a new equation without the \(xy\)-term, making it easier to understand and graph the conic shape. This refinement simplifies analysis and sketching, allowing for a clearer comprehension of the conic's structure. By using rotation, complex relationships transform into more recognizable forms.