Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{x^{2}}{9}-\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The conic is a horizontal hyperbola.

Step 1: Identify the Standard Form

The given equation is \( \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1 \). This is in the standard form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), which represents a hyperbola.

Step 2: Determine the Orientation

In the standard form of a hyperbola \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), if the \( x^{2} \) term is positive and \( y^{2} \) term is negative, it indicates a horizontal hyperbola.

Key Concepts

Hyperbolas

A hyperbola is a fascinating type of conic section formed by the intersection of a plane with a double-napped cone. Unlike other conics, such as ellipses and parabolas, hyperbolas consist of two separate curves known as branches. These branches mirror each other and extend infinitely in opposite directions.

Hyperbolas have distinctive properties:

• Two foci (plural of focus) and two directrices.
• Two asymptotes that intersect at a point known as the center of the hyperbola. These asymptotes act as guidelines for the branches, becoming closer to them as the branches extend.
• The line segment passing through the foci is called the transverse axis, which is an essential feature. It's the axis where the branches open.
• Additionally, there's a conjugate axis that is perpendicular to the transverse axis.

Understanding these properties helps in sketching hyperbolas and deciphering their algebraic equations.

Standard Form of Conics

The standard form of a conic is an equation used to express different types of conic sections, such as circles, ellipses, parabolas, and hyperbolas in a simplified manner. When dealing with conics, recognizing their standard forms is crucial for understanding and analyzing them.

For hyperbolas, the standard form is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]The positive term's variable indicates the axis along which the hyperbola is oriented.

• If the term \( \frac{x^2}{a^2} \) is positive, it's a horizontal hyperbola.
• If the term \( \frac{y^2}{b^2} \) is positive, it's a vertical hyperbola.

Each conic section's standard form allows for easy classification and helps in the identification of specific features relevant to that conic.

Equation of a Hyperbola

The equation of a hyperbola is essential for its identification and analysis. As part of the family of conic sections, hyperbolas have distinct forms.
The standard equation: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]is used to represent a hyperbola centered at the origin with the transverse axis along the x-axis.
In this equation, \(a\) and \(b\) denote lengths associated with the hyperbola:

• \(a\) is the distance from the center to each vertex on the transverse axis.
• \(b\) is related to the distance from the center to points on the conjugate axis.

The hyperbola equation can reveal the orientation (horizontal or vertical), center, foci, and vertices of the hyperbola.

• For a hyperbola centered at the origin, its foci are at points \((\pm c, 0)\) or \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\).
• Vertices occur at \((\pm a, 0)\) or \((0, \pm a)\), confirming the direction in which the hyperbola opens.

Recognizing and using these parameters effectively aids in graphing the hyperbola and solving related problems.