Problem 3 - Intermediate Algebra

Question

Identify the graph of each equation as an ellipse or a hyperbola. Do not graph. \(x^{2}-5 y^{2}=3\)

Step-by-Step Solution

Answer

This is a hyperbola.

Step 1: Identify the Standard Form

The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Our equation is \(x^2 - 5y^2 = 3\), which resembles the form of a hyperbola equation.

Step 2: Rewrite the Equation in Standard Form

Divide the entire equation \(x^2 - 5y^2 = 3\) by 3 to get \(\frac{x^2}{3} - \frac{5y^2}{3} = 1\). This matches the standard form of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 3: Conclude the Type of Graph

Since the equation \(\frac{x^2}{3} - \frac{5y^2}{3} = 1\) takes the form of a hyperbola, where one term is subtracted from the other, it is an equation of a hyperbola.

Key Concepts

Standard Forms

In mathematics, the standard forms of conic sections are vital in recognizing and distinguishing different shapes like circles, ellipses, parabolas, and hyperbolas. Understanding these forms helps to identify the equation's graph type without even plotting the graph.
For hyperbolas, the standard forms are:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

These forms show that either the term with \(x^2\) or \(y^2\) is subtracted, indicating the distinctive shape of hyperbolas.
On the other hand, ellipses have the following standard form:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

This form indicates that both terms are added, representing an enclosed symmetric shape. By rewriting an equation into one of these standard forms, you can easily determine the conic section type without graphing it, as shown with the rewritten equation \( \frac{x^2}{3} - \frac{5y^2}{3} = 1 \), where the subtraction identifies it as a hyperbola.

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane at different angles. Depending on how the plane intersects the cone, different shapes are formed, known as conic sections. These include:

• Circles: When the plane is perpendicular to the cone's axis.
• Ellipses: When the plane cuts the cone at an angle, but doesn’t go through the base.
• Parabolas: Formed when the plane is parallel to a tangent line of the cone.
• Hyperbolas: Occur when the plane cuts both halves of the cone.

Studying these shapes is critical in mathematics due to their diverse properties and applications. Each conic section has its unique standard form, which helps in identifying and analyzing them algebraically. For instance, the original equation \(x^2 - 5y^2 = 3\) was recognized as a hyperbola by converting it into its standard form. Understanding these sections aids in solving complex geometric and algebraic problems.

Graphing Equations

Graphing conic sections involves plotting them based on their standard forms. However, recognizing them algebraically beforehand eases this process, as seen in our exercise. To graph:

• Rewrite the equation into the standard form to understand its shape.
• Determine key features: center, vertices, foci, asymptotes (for hyperbolas), major and minor axes (for ellipses).
• Plot these features on a coordinate plane.

For hyperbolas such as \( \frac{x^2}{3} - \frac{5y^2}{3} = 1 \), you would locate the center and the asymptotes, which guide the hyperbola's opening direction. Asymptotes for hyperbolas help in drawing their two separate curves.
Through this approach, you develop a clearer understanding of the graph's layout and can accurately represent the equation visually, facilitating deeper comprehension of the equation's properties and behavior.