Problem 83 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that depending on the values for \(A\) and \(B\), assuming that they are both not zero, the graph of \(A x^{2}+B y^{2}=C\) can represent any of the conic sections other than a parabola.

Step-by-Step Solution

Answer

The statement is sensible and accurate given the algebraic representations of various conic sections.

Step 1: Identifying Conic Sections

There are primarily four types of conic sections - circle, ellipse, hyperbola, and parabola. The given equation \(Ax^2 + By^2 = C\) does not cover all these types. Here's why: circle and ellipse are represented by equations of type \(Ax^2 + By^2 = C\), where A = B (for a circle) and A ≠ B (for an ellipse). A hyperbola is represented by an equation of the type \(Ax^2 - By^2 = C\). As can be seen, these are covered by the given equation. However, a parabola is represented by either \(y^2 = 4ax\) or \(x^2 = 4ay\), which is evidently not covered by the given equation.

Step 2: Evaluating the Statement

The statement suggests that the equation \(Ax^2 + By^2 = C\) can represent any conic section except a parabola, which is indeed correct as we found in the previous step.

Step 3: Concluding the Statement

It can be concluded that the statement 'depending on the values for A and B, assuming that they are both not zero, the graph of \(Ax^2 + By^2 = C\) can represent any of the conic sections other than a parabola' makes sense and is accurate according to the algebraic representations of conic sections.

Key Concepts

Circle

When we talk about a circle in mathematics, we are referring to a set of points in a plane. These points are equidistant from a central point known as the center. This distance is called the radius.

A circle has a very specific equation:

• The standard form is \(x^2 + y^2 = r^2\), where \r\ is the radius.
• In an expanded form, with both \(A\) and \(B\) being equal as in \(Ax^2 + By^2 = C\), a circle is formed when \(A = B\) and is positive.

The reason circles fall under the category of conic sections is because they can be formed by cutting a cone parallel to its base.
Circles are among the simplest conic sections to understand and have symmetric properties, which make them quite unique.

Ellipse

An ellipse is another interesting form of conic section, with properties that are similar to a circle but stretched in one direction. Unlike circles, ellipses have two axes: a major axis and a minor axis.

The standard equation for an ellipse is more flexible:

• It can be expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• When \(A eq B\), the equation \(Ax^2 + By^2 = C\) illustrates an ellipse as long as both coefficients are positive.

Ellipses appear when a plane cuts through a cone at an angle, not parallel or perpendicular to the base.
They have the remarkable property of stretching into ovals, but fundamentally they maintain a consistent mathematical relationship between dimensions. This axis-based deformation gives them distinct characteristics.

Hyperbola

A hyperbola is a type of conic section that forms an open curve, which looks like two mirrored parabolas back-to-back. It's different from circles and ellipses due to how it behaves asymptotically.

The hyperbola's standard form equation is:

• \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• For the equation \(Ax^2 - By^2 = C\), a hyperbola is formed when one coefficient is positive and the other is negative.

Hyperbolas appear when a plane intersects both halves of a double cone, resulting in their distinctive twin-curve shapes.
The hyperbola's unique properties come from the hyperbolic nature of distances and the pathways that continuously diverge from each other, unlike ellipses and circles.

Parabola

A parabola is perhaps the most distinct conic section because of its simple shape and relationship with quadratic functions. A parabola can be recognized easily by its "U" shape, opening either up, down, right, or left.

Its standard form is different from others:

• One common form is \(y^2 = 4ax\) for a parabola opening along the x-axis.
• Another is \(x^2 = 4ay\) for one opening in the y-axis direction.

Parabolas occur when a plane cuts a cone parallel to one of its sides, which leads to its specific open curve always being completed.
This separation from the other conic sections demonstrates how they cannot be formed by the equation \(Ax^2 + By^2 = C\), which requires both higher powers of terms or both terms to be squared in distinction.