Problem 2 - College Algebra with Corequisite Support

Question

If the equation of a conic section is written in the form \(A x^{2}+B y^{2}+C x+D y+E=0\) and \(A B=0\), what can we conclude?

Step-by-Step Solution

Answer

The equation is a parabola.

Step 1: Identify the Types of Conic Sections

The general equation of a conic section is given by \(A x^{2} + B y^{2} + C x + D y + E = 0\). The types of conic sections include parabolas, ellipses, circles, and hyperbolas. The coefficients \(A\) and \(B\) help to determine the type of conic section.

Step 2: Conditions for Parabolas

For the conic to be a parabola, either \(A = 0\) or \(B = 0\), but not both. This means the equation will have a single squared term, either \(x^2\) or \(y^2\).

Key Concepts

Parabolas

A parabola is a unique type of conic section that can be easily identified by its distinct shape, resembling a U or an inverted U. It forms when one's plane slices through a cone parallel to its side. Parabolas have a range of practical applications, from satellite dishes to the paths of projectiles.

Mathematically, a parabola can be expressed in the form of the equation: \[ y^2 = 4ax \] (if the parabola opens horizontally) or \[ x^2 = 4ay \] (if it opens vertically).
Some key features include:

• Vertex: The point where the parabola changes direction.
• Axis of Symmetry: A line that divides the parabola into two mirror images.
• Focus: A point at which all paths reflected from the parabola converge.

In the context of the conic sections equation \(Ax^2 + By^2 + Cx + Dy + E = 0\), a parabola occurs when either \(A = 0\) or \(B = 0\), but not both.
This results in only one of the variables having a squared term, which is characteristic of parabolas.

Equation of Conic Sections

The equation of a conic section is a mathematical expression that describes the various shapes obtained when a plane intersects a cone. These equations are crucial for understanding and classifying different conics such as circles, ellipses, parabolas, and hyperbolas. The general equation is represented as:\[ Ax^2 + By^2 + Cx + Dy + E = 0 \]
Here, the coefficients \(A\), \(B\), \(C\), \(D\), and \(E\) play pivotal roles in determining the shape of the conic section.

• Coefficients \(A\) and \(B\): These determine the type of conic section. If \(AB = 0\), we conclude that the section is a parabola, as there is only one squared term.
• Other coefficients: \(C\) and \(D\) influence the position, while \(E\) contributes to the equation's overall shape and location.

Understanding this equation helps in graphing the conics and predicting their shapes based on the values of these coefficients.

Types of Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The positioning of the plane defines the various types of conic sections: ellipse, circle, parabola, and hyperbola. Each type has its unique properties and can be derived by adjusting the general conic section equation parameters.

Here are the types with their characteristics:

• Ellipses: This occurs when the plane cuts through the cone at an angle to its base, but does not intersect the base. Their equation has both \(x^2\) and \(y^2\) with the same sign.
• Circles: A special type of ellipse where \(A = B\). It appears as a perfect round shape.
• Parabolas: Arise when \(A = 0\) or \(B = 0\), but not both, leaving one squared term in the equation.
• Hyperbolas: Form when the plane intersects both nappes of the cone, and the equation has \(x^2\) and \(y^2\) with opposite signs.

By adjusting and analyzing these parameters, each conic section can be distinctly classified, offering insight into the shape and properties of the curve.