Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

It's a degenerate hyperbola.

Step 1: Identify the equation format

We start by identifying the given equation: \( \frac{-x^2}{9} = \frac{y}{4} \). This needs to be compared to the general forms of conic sections equations.

Step 2: Rewrite the equation

First, rewrite the equation so that all terms are on one side: \( \frac{-x^2}{9} - \frac{y}{4} = 0 \). Multiply through by a common denominator, if necessary.

Step 3: Recognize the conic section

The key feature is the negative sign on the \(x^2\) term, which is \( \frac{-x^2}{9} - \frac{y}{4} = 0 \). It suggests a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) (notice the signs are opposite).

Step 4: Consider the form

Given equation does not resemble directly the standard hyperbola form but can be transformed by multiplying by -1 and rearranging: \( \frac{x^2}{9} + \frac{y}{4} = 0 \) suggests a hyperbola but won't equate neatly to 1, showing degeneracy. This confirms it is a degenerate hyperbola.

Key Concepts

Understanding Hyperbolas

Hyperbolas are one of the fascinating shapes in conic sections that arise from the intersection of a plane with a double cone. They consist of two opposite-facing arms that mirror each other. Unlike ellipses or circles, hyperbolas occur when the plane cuts through both halves of the double cone.

• Standard Form: A typical form of the hyperbola equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where the variables \( x \) and \( y \) have opposite signs.
• Characteristics: The curvature and orientation depend on the values of \( a \) and \( b \); the larger these values, the wider the hyperbola.

The key here is the negative sign of one of the squared terms, indicating the unique interaction that forms this conic section. Some hyperbolas also open side-to-side, others open up-and-down, based on how the positive and negative terms are presented.

Exploring Ellipses

An ellipse forms a closed loop and represents yet another distinct conic section. It resembles a stretched circle, created by a plane's unique angle slicing through a single cone.

• Standard Form: The equation for an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• Different Axes: If both \( a \) and \( b \) are equal, the ellipse is a circle. If not, the ellipse is longer along one axis.

Ellipses have various real-world applications, like planetary orbits, where their balanced symmetry supports consistent rotation and revolution patterns.

Recognizing Equation Forms

Understanding the equation forms of conic sections aids in identifying their types. Conic sections include shapes derived from quadratic relations between two variables.

• General Form: A quadratic equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) may represent any conic section.
• Signs and Values: The coefficients \( A, B, \) and \( C \) determine whether an equation signifies a circle, ellipse, parabola, or hyperbola.

The intriguing aspect of studying these equations is identifying the nuances like signs and coefficients that distinguish one conic from another. This subtle difference, for example, turns an elliptical equation into a hyperbola, making problem-solving a creative exercise.

Degenerate Conics Explained

Degenerate conics are special cases of conic sections where the conic degenerates into simpler forms. These results occur when specific parameters in the conic's equation satisfy certain conditions, leading to simpler outputs.

• Simple Forms: Degenerate forms include intersecting lines, a single point, or even no real intersection at all.
• Example in Context: In the exercise, the equation \( \frac{x^2}{9} + \frac{y}{4} = 0 \) hinted at a hyperbola, but since there's no real solution (i.e., it doesn't equate to 1), it's a degenerate case.

Identifying degenerate conics requires careful attention when comparing an equation’s standard form statements. This nature of equations means they have evolved enough within their expressions to no longer present as typical curves.