Problem 59 - Intermediate Algebra

Question

If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their equations.

Step-by-Step Solution

Answer

Different conics are identified by comparing coefficients in the equation: circles have \( A = C \), parabolas have either \( A \) or \( C \) as zero, ellipses have \( A \) and \( C \) same sign but unequal, hyperbolas have opposite signs for \( A \) and \( C \).

Step 1: Identify the General Form

Conic sections are mathematical curves that can be represented by a general second-degree polynomial equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). This equation can represent a circle, parabola, ellipse, or hyperbola depending on the coefficients \( A, B, \) and \( C \).

Step 2: Distinguish a Circle

For a circle, the equation will have the form \( x^2 + y^2 + Dx + Ey + F = 0 \). In this case, the coefficients \( A \) and \( C \) are equal and the coefficient \( B \) is zero. This means there is no cross-product term \( xy \), and the coefficients of \( x^2 \) and \( y^2 \) are the same.

Step 3: Identify a Parabola

A parabola can be identified if either \( A \) or \( C \) is zero, but not both. This means there is no term of one of the squared variables \( x^2 \) or \( y^2 \). For vertical parabolas, the equation typically looks like \( y = ax^2 + bx + c \), and for horizontal parabolas like \( x = ay^2 + by + c \).

Step 4: Recognize an Ellipse

An ellipse will have the form \( Ax^2 + Cy^2 + Dx + Ey + F = 0 \) with \( A \) and \( C \) having the same sign but not equal, and \( B \) is zero. If there is no rotation, there is no \( xy \) term, and \( A eq C \).

Step 5: Distinguish a Hyperbola

A hyperbola is identified by the form \( Ax^2 + Cy^2 + Dx + Ey + F = 0 \) where \( A \) and \( C \) have opposite signs. There is no restriction on the \( B \) coefficient being zero or not if the hyperbola is rotated.

Key Concepts

General Form of Conic Sections

Conic sections are fascinating curves that arise when a plane intersects a double-napped cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each conic is unique but can be described by a similar general quadratic equation: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] This equation involves six coefficients: \( A, B, C, D, E, \) and \( F \).

• The presence and values of these coefficients determine which type of conic section the equation represents.
• Understanding how these coefficients interact is essential to classifying the conic.
• The terms \( x^2 \) and \( y^2 \) are crucial in determining the shape of the conic.

The simplified general form helps in quickly identifying circles, ellipses, parabolas, and hyperbolas through a few key observations.

Equations of Circles

To identify a circle from its equation, note that both \( x^2 \) and \( y^2 \) terms are present and have the same coefficients in the general form: \[ x^2 + y^2 + Dx + Ey + F = 0 \] Important characteristics of a circle's equation include:

• The coefficients \( A \) and \( C \) are equal, ensuring the circular symmetry in both the \( x \) and \( y \) directions.
• There is no cross-product term, which means \( B = 0 \). This absence of an \( xy \) term implies no rotation.

Understanding these features helps quickly pinpoint a circle among other conic sections.

Equations of Parabolas

Parabolas are unique among conics because they result from aligning constants \( A \) or \( C \) to zero. For parabolas:

• Either \( A = 0 \) or \( C = 0 \), but not both. This removal of either the \( x^2 \) or \( y^2 \) term ensures a parabolic structure.
• Vertical parabolas have the form \( y = ax^2 + bx + c \), emphasizing the dominant role of the \( x \) variable.
• Horizontal parabolas showcase the structure \( x = ay^2 + by + c \), where \( y \) plays the primary role in the equation.

These forms help distinguish parabolas and highlight their distinct symmetrical properties along a single axis.

Equations of Ellipses

Ellipses are slightly elongated circles, represented when the coefficients \( A \) and \( C \) have the same sign but different values in the equation:\[ Ax^2 + Cy^2 + Dx + Ey + F = 0 \] Recognizable features of ellipse equations include:

• Both \( A \) and \( C \) must be positive or both negative, but \( A eq C \). This indicates the ellongation in one direction.
• No cross-product terms present, thus \( B = 0 \), indicating no rotation, maintaining the basic orientation.

Ellipses share properties with circles but stretch more along one axis, leading to their distinct shape.

Equations of Hyperbolas

Hyperbolas are characterized by the opposite signs of \( A \) and \( C \) in the equation:\[ Ax^2 + Cy^2 + Dx + Ey + F = 0 \] Key features of hyperbolas include:

• The distinguishing factor is \( A \) and \( C \) having opposite signs, like \( A \) positive and \( C \) negative, or vice versa.
• Unlike ellipses and circles, there may be an \( xy \) term when the hyperbola is rotated, which affects orientation.

This makes hyperbolas visually and algebraically unique, creating the shape of two separate curves or branches.