Problem 3 - College Algebra

Question

If the equation of a conic section is written in the form \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) , and \(B^{2}-4 A C > 0\) , what can we conclude?

Step-by-Step Solution

Answer

The conic section is a hyperbola.

Step 1: Understand the Equation

The given equation is a general form of a conic section: \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\). This can represent different conic sections: circles, ellipses, parabolas, and hyperbolas, depending on the coefficients.

Step 2: Analyze the Discriminant

The discriminant for the conic section is given by \(B^{2}-4AC\). This value determines the type of conic. If \(B^{2}-4AC > 0\), it indicates the nature of the conic section.

Step 3: Identify the Conic Section

When the discriminant \(B^{2}-4AC > 0\), it specifies that the equation represents a hyperbola. This is because, for hyperbolas, the condition \(B^{2}-4AC > 0\) is satisfied.

Key Concepts

Equation of Conic Section

Conic sections can be described by a general second-degree polynomial equation:\[A x^{2} + B x y + C y^{2} + D x + E y + F = 0\]This equation can represent different shapes known as conic sections. These shapes include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These conic sections are defined according to the values of the coefficients \(A\), \(B\), and \(C\).
Understanding the roles of these coefficients is crucial in identifying the type of conic section from a given equation. The equation provides a unified way of analyzing these different geometric shapes.

Discriminant

The discriminant in the context of conic sections is a valuable tool. It is calculated using the formula:\[B^{2} - 4AC\]The discriminant helps us determine the type of conic section represented by the equation. Depending on its value, we can understand which conic section we are dealing with:

• If \(B^{2} - 4AC > 0\), the equation represents a hyperbola.
• If \(B^{2} - 4AC = 0\), it represents a parabola.
• If \(B^{2} - 4AC < 0\), it shows the equation is that of an ellipse or a circle.

Each of these conditions corresponds to a particular geometric shape. The discriminant serves as a shortcut to identify the nature of the curve without needing to graph it.

Hyperbola

A hyperbola is a type of conic section that appears in two branches, which are mirror images of each other.For an equation of the form \(A x^{2} + B x y + C y^{2} + D x + E y + F = 0\), if \(B^{2} - 4AC > 0\), we indeed have a hyperbola.Here are some characteristics of hyperbolas:

• They have two distinct open curves.
• Each branch tends to infinity.
• The branches are symmetrical with respect to each other.
• They can be represented in different orientations, either opening horizontally or vertically.

Understanding hyperbolas involves grasping these properties and being able to define them through their standard equations.

Types of Conic Sections

Conic sections come in four primary types, each characterized by specific properties:

• Circle: A special type of ellipse where \(A = C\) and \(B = 0\). All points are equidistant from a central point.
• Ellipse: An elongated circle, formed when \(B^{2} - 4AC < 0\) and \(A eq C\). Its shape is defined by its major and minor axes.
• Parabola: Illustrates the path of a projectile; identified when \(B^{2} - 4AC = 0\).
• Hyperbola: Consists of two branches, occurring when \(B^{2} - 4AC > 0\).

These four types demonstrate the variety and beauty of conic sections. Each type can be recognized by its unique equation format, providing insight into its geometric structure.