Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, determine which conic section is represented based on the given equation. \(-3 x^{2}+3 \sqrt{3} x y-4 y^{2}+9=0\)

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the General Form of a Conic Section

First, let's rewrite the given equation in a more standard form for conic sections. The general form of a conic section is \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]. Our equation is \[ -3x^2 + 3\sqrt{3}xy - 4y^2 + 9 = 0 \], which already fits this form with \( A = -3 \), \( B = 3\sqrt{3} \), \( C = -4 \), \( D = 0 \), \( E = 0 \), and \( F = 9 \).

Step 2: Calculate the Discriminant to Classify the Conic

The discriminant \( \Delta \) of a conic section helps determine the type of conic section represented by the equation:\[ \Delta = B^2 - 4AC \].Substitute the values:\[ B = 3\sqrt{3}, \quad A = -3, \quad C = -4 \].Calculate \( \Delta \):\[ \Delta = (3\sqrt{3})^2 - 4(-3)(-4) = 27 - 48 = -21 \].

Step 3: Determine the Conic Section Type Using the Discriminant

The sign of the discriminant indicates the conic section:- If \( \Delta < 0 \), the conic section is an ellipse.- If \( \Delta = 0 \), the conic section is a parabola.- If \( \Delta > 0 \), the conic section is a hyperbola.In our case, since \( \Delta = -21 \), which is less than zero, the conic section represented by the equation is an ellipse.

Key Concepts

Ellipse

An ellipse is one of the fundamental shapes in conic sections, resembling an elongated circle or an oval. Imagine stretching a circle outwards while keeping its overall symmetry; that's essentially what an ellipse is. Every ellipse has two focal points, commonly called foci, and it's the sum of the distances from any point on the ellipse to these two foci that remains constant. This unique property distinguishes ellipses from other conic sections.

Ellipses are prevalent in both mathematics and the natural world. For example:

• The orbits of planets around the sun in our solar system are elliptical.
• Ellipses are used in engineering and design due to their pleasing aesthetic properties.
• In acoustics, ellipses can predict how sound waves behave in certain spaces.

Understanding ellipses can help us comprehend various physical and abstract phenomena, making them a crucial concept in mathematics and science.

Discriminant

The discriminant of a conic section is a mathematical expression used to help classify the conic's type based on the coefficients of its general equation. It provides a quick way to determine whether the conic section is a parabola, ellipse, or hyperbola. The formula for the discriminant of a conic section is:\[ \Delta = B^2 - 4AC \]

The variables \(A\), \(B\), and \(C\) are coefficients taken from the general equation of a conic section, which is expressed as:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

By calculating \(\Delta\), we get:

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

In the provided exercise, the discriminant is calculated to be \(-21\), which is negative, accurately indicating that the conic section is an ellipse. This calculation is straightforward and showcases the power of discriminants in quickly categorizing conic sections.

Conic Section Classification

Conic section classification involves identifying the geometric shape that is represented by a quadratic equation in two variables. The four primary types of conic sections are the circle, ellipse, parabola, and hyperbola. Each type has its own characteristics and can be recognized by analyzing the equation's coefficients and discriminant.

Classification typically follows these steps:

• Identify the general form of the conic equation.
• Substitute the coefficients into the discriminant formula: \( \Delta = B^2 - 4AC \).
• Based on the value of \( \Delta \), classify the conic section.

This classification is not just theoretical. It has practical applications in astronomy, physics, and engineering. For instance:

• In architecture, ellipses may be used to design archways and domes.
• Parabolas are used in reflectors, such as satellite dishes and flashlight mirrors.
• Hyperbolas can describe certain types of orbits and are related to phenomena in relativity.

The knowledge of classifying conic sections is pivotal in many scientific fields, providing clues about the inherent properties of different spaces and curves.