Problem 39 - Thomas Calculus

Question

Can anything be said about the graph of the equation \(A x^{2}+B x y+\) \(C y^{2}+D x+E y+F=0\) if \(A C<0 ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

The graph is a hyperbola since \(AC < 0\) implies \(B^2 - 4AC > 0\).

Step 1: Identify Conic Section Equation

The given equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) is the general form of a conic section. Conic sections can be circles, ellipses, parabolas, or hyperbolas, depending on the relationship between the coefficients \(A\), \(B\), and \(C\).

Step 2: Consider Condition for Hyperbola

To determine the type of conic section, we use the discriminant \(B^{2} - 4AC\). If \(B^2 - 4AC > 0\), the conic is a hyperbola. The condition \(AC < 0\) implies that the product of \(A\) and \(C\) is negative, which means \(B^2 - 4AC > 0\) is always satisfied.

Step 3: Conclusion on Conic Type

Given \(AC < 0\), it ensures that \(B^2 - 4AC > 0\). Therefore, the conic section represented by the equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) is a hyperbola.

Key Concepts

Hyperbola

In the vast universe of conic sections, the hyperbola holds a unique position. A hyperbola is formed by the intersection of a double cone with a plane that cuts through both nappes (the top and bottom parts) of the cone. Unlike ellipses and parabolas, hyperbolas have two distinct branches which mirror each other.
Hyperbolas are defined by their eccentricity, which is always greater than one. This characteristic shapes their wide, open curves. In terms of equation, a hyperbola in its standard form is expressed as:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] with the transverse axis along the x-direction, or:
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]with the transverse axis along the y-direction.Common features of hyperbolas include:

• Two disjoint branches
• Asymptotes that guide the shape of the curve
• A center point, which is not a point on the hyperbola itself
• Vertices that mark the closest points on each branch

Recognizing a hyperbola in a conic equation where the general form is used, like the one in the problem, often involves checking the condition of the coefficients as explored with the discriminant.

Discriminant of Conic Sections

The discriminant is a powerful tool to deduce the type of conic section represented by a general quadratic equation. The discriminant of a conic section is given by the formula:
\[B^2 - 4AC\]It helps in identifying the nature of the graph of the equation.
Here's how it works:

• If \(B^2 - 4AC > 0\), the conic section is a hyperbola.
• If \(B^2 - 4AC = 0\), it represents a parabola.
• If \(B^2 - 4AC < 0\), then the conic section could be a circle or an ellipse.

In our specific problem, since \(AC < 0\), this guarantees that \(B^2 - 4AC > 0\), leading to the conclusion that the conic denoted by the given equation is invariably a hyperbola.
Understanding this discriminant criterion is key to solving quadratic equations in two variables because it simplifies the task of categorizing conic sections, making it a fundamental concept in algebraic geometry.

General Form of Conic Sections

The general form of a conic section is a versatile representation that can describe various shapes such as circles, ellipses, parabolas, and hyperbolas, within a single formula:
\[A x^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]Here's what each term represents:

• \(A x^2\), \(Bxy\), and \(Cy^2\) are responsible for the actual conic shape. They are quadratic terms that define the nature of the conic.
• \(Dx\) and \(Ey\) are linear terms that can shift the conic section within the coordinate plane.
• \(F\) is the constant term that moves the graph up or down.

Depending on the values of \(A\), \(B\), and \(C\), the type of conic section changes, evaluated effectively using the discriminant.
This form enables us to switch between different types of conic sections by only changing the coefficients, highlighting the symmetry and interconnection between these geometric figures. It is the underlying structure that can be specialized into particular cases, like the standard forms of hyperbolas, ellipses, and parabolas, depicting their geometry easily.