Problem 8 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$2 x^{2}-2 y^{2}+4 x-6 y-2=0$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the General Form

The given equation is in the form:\[ ax^2 + by^2 + cx + dy + e = 0 \]where the coefficients are: \( a = 2 \), \( b = -2 \), \( c = 4 \), \( d = -6 \), and \( e = -2 \).

Step 2: Compare Coefficients of Variables

For a conic section equation in the general form \( ax^2 + by^2 + cx + dy + e = 0 \), the type of conic section depends on the relationship between \( a \) and \( b \):- If \( a = b \), it is a circle.- If \( ab = 0 \) (one coefficient is zero), it is a parabola.- If \( ab > 0 \) (same sign), it is an ellipse.- If \( ab < 0 \) (different signs), it is a hyperbola.Here, \( a = 2 \) and \( b = -2 \), so \( ab = 2(-2) = -4 \). Since \( ab < 0 \), the conic section is a hyperbola.

Step 3: Conclusion

Based on the calculation in Step 2 where \( ab < 0 \), we conclude that the given equation represents a hyperbola.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola's equation can vary depending on its orientation. There are two main forms:

• Horizontal hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
• Vertical hyperbola: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

Here, \((h, k)\) is the center of the hyperbola, and \(a\) and \(b\) are the distances from the center to the vertices and co-vertices, respectively.
Hyperbolas have distinctive properties, such as asymptotes, which are lines that the hyperbola approaches but never touches. This makes their overall shape similar to two opposing arcs. Understanding the hyperbola helps in identifying it from its equation by comparing the coefficients, as we can see in the given problem.

General Form of a Conic

The general form of a conic section equation is expressed as:\[ ax^2 + by^2 + cx + dy + e = 0 \]This equation encompasses all conic sections, including circles, ellipses, parabolas, and hyperbolas. Each type of conic has its own characteristic formed by the values and interplay of the coefficients \(a\), \(b\), \(c\), \(d\), and \(e\).
In the given exercise, the equation \(2x^2 - 2y^2 + 4x - 6y - 2 = 0\) is already in general form. By analyzing the coefficients, we can deduce the type of the conic section. Each coefficient plays a role in shaping the curve, making this form a powerful tool in identifying conics.

Comparison of Coefficients

Comparing the coefficients in a conic's general equation is crucial for identifying the type of conic section. The relationship between the coefficients \(a\) and \(b\) reveals a lot about the conic's nature:

• If \(a = b\), it's a circle, symmetrical in all directions.
• If one of \(a\) or \(b\) is zero, the equation represents a parabola, indicating a single, open curve.
• If \(ab > 0\), both having the same sign, it's an ellipse, which includes a circle as a special case.
• If \(ab < 0\), meaning they have opposite signs, it's a hyperbola, which we saw was the case in the exercise.

Here, by analyzing that \(a = 2\) and \(b = -2\), leading to \(ab = -4\), the equation was identified as representing a hyperbola.

Types of Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double right circular cone. There are four primary types:

• Circle: A special type of ellipse where \(a = b\).
• Ellipse: Defined by \(ab > 0\), indicating an oval shape.
• Parabola: Formed when either \(a = 0\) or \(b = 0\), leading to a single curve opening in one direction.
• Hyperbola: Identified by \(ab < 0\), characterized by two branches opening in opposite directions.

Each type has unique properties and equations that define its shape and orientation. Understanding these differences helps in accurately interpreting and analyzing mathematical problems involving conic sections, such as the one provided in the exercise.