Problem 3 - College Algebra with Corequisite Support

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 equation represents a hyperbola.

Step 1: Identify the Type of Conic Section Criteria

In conic sections, the discriminant \( B^2 - 4AC \) helps in identifying the type of conic represented by the general equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

Step 2: Apply the Discriminant Condition

The condition \(B^2 - 4AC > 0\) indicates that the conic section is a hyperbola. This is because a hyperbola is defined as having a positive discriminant.

Key Concepts

Discriminant in Conics

The discriminant in conics is a crucial tool to determine the type of conic section represented by a general quadratic equation. The equation is given by \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]where A, B, and C are coefficients. The discriminant is calculated using the formula \[ B^2 - 4AC \].Its value tells us the nature of the conic:

• If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), the conic is a parabola.
• If \( B^2 - 4AC < 0 \), the conic is an ellipse (or circle, if \( A = C \)).

Understanding this discriminant not only identifies the conic type but also helps in sketching these curves. This concept provides a mathematical way to classify shapes that arise in various physical and mathematical problems.

Hyperbola

A hyperbola is a distinctive conic section characterized by a positive discriminant \( B^2 - 4AC > 0 \). It is generally formed when the intersection of a double cone and a plane results in two separate, mirror-image curves. Some key features of hyperbolas include:

• Branches: They have two distinct branches.
• Standard Form: Can be represented by \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]for horizontally oriented, or \[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]for vertically oriented hyperbolas.
• Asymptotes: They approach two asymptotes that intersect at the center of the hyperbola.
• Foci: Located along the transverse axis inside each branch.

Hyperbolas have intriguing applications, which can range from physics—like hyperbolic trajectories of celestial bodies—to engineering, where antenna designs often use them for efficiency.

General Equation of a Conic

The general equation of a conic encompasses all conic sections such as parabolas, ellipses, circles, and hyperbolas. Written as \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \],each conic section is determined based on the values of the coefficients A, B, C, D, E, and F. This equation presents a comprehensive form that can model any conic shape:

• Parabola: Results when the discriminant \( B^2 - 4AC = 0 \).
• Ellipse (or Circle): Occurs when \( B^2 - 4AC < 0 \).
• Hyperbola: Arises when \( B^2 - 4AC > 0 \), illustrating the separateness of the curve's branches.

This versatile approach allows one to algebraically express and transition between different types of conic sections with ease by modifying the coefficients. It underlines the relationship between algebra and geometry, showing how equations correspond to geometric shapes.