Problem 4 - 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 of the Equation

Examine the given equation \( \frac{-x^{2}}{9} + \frac{y^{2}}{4} = -1 \). Notice that it fits the standard form of a hyperbola, which is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). In this equation, the terms are switched but preserves the form of a hyperbola \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = -1 \).

Step 2: Change the Equation to a Recognizable Standard Form

We need to match the structure of a standard hyperbola. Multiply the entire equation by -1 to match the standard form: \( \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1 \).

Step 3: Analyze the Leads of the Hyperbola

Observe that \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \) begins with the \( x^2 \) term as positive. In the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the x-term being positive indicates the hyperbola is horizontal.

Key Concepts

Conic Sections

Conic sections are fascinating curves that can be formed by the intersection of a plane and a cone. The angle and position of the intersection determine the type of conic section. There are four main types of conic sections:

• Circle: Formed when the intersecting plane is perpendicular to the cone’s axis.
• Ellipse: A closed curve, appearing when the intersecting angle is less than the angle of the cone.
• Parabola: An open curve formed when the plane is parallel to one of the cone’s slant lines.
• Hyperbola: Created when the plane intersects both halves of the cone, resulting in two separate curves.

Each conic section has its unique properties and equations that describe its geometric nature. Hyperbolas are particularly interesting because they consist of two disconnected curves, called branches, and they have a central axis.

Standard Form of Conic Equations

The equations of conic sections can be expressed in a standard mathematical form which helps in identifying and analyzing their properties. For hyperbolas, the standard form of the equation typically looks like:

• For horizontal hyperbolas: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• For vertical hyperbolas: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

In these expressions, the values \( a \) and \( b \) determine the shape of the hyperbola, specifying the distance from the center to the vertices and the slopes of the asymptotes, respectively.

Recognizing this standard form makes it easier to figure out the orientation of the hyperbola. The sign before the \( x^2 \) or \( y^2 \) term tells you if the hyperbola is horizontal or vertical. For circles and ellipses, the standard form is different, highlighting how distinct each conic section's equation can be.

Equations of Hyperbolas

Hyperbolas are a type of conic section that can be identified by their unique equation forms. Their equations are identifiable due to their specific structure, featuring subtraction between the squared terms:

• A typical hyperbola is written as: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
• The equation can also appear in a slightly altered form, where subtracting changes the arrangement: \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = -1 \).

The negative sign between the terms is the hallmark of hyperbolas, distinguishing them from ellipses, which use addition.

Modifying these equations can help convert them back to their standard form for easier analysis. A small manipulation, like multiplying the entire equation by \(-1\), can clarify whether the hyperbola is horizontal or vertical. In the given exercise, this transformation reveals how the hyperbola opens and its orientation.