Degenerate Hyperbola

Definition

A degenerate hyperbola is a special case of a hyperbola where the two branches of the hyperbola coincide, forming a single straight line. This occurs when the coefficients of the hyperbola equation are such that the hyperbola degenerates into a line segment.

5 Must Know Facts For Your Next Test

1. A degenerate hyperbola occurs when the coefficients of the hyperbola equation are such that the discriminant $b^2 - a^2$ is equal to 0.
2. In a degenerate hyperbola, the two branches of the hyperbola coincide, forming a single straight line.
3. The equation of a degenerate hyperbola can be written in the form $ax^2 + 2bxy + cy^2 = 0$, where $b^2 - ac = 0$.
4. Degenerate hyperbolas have no interior points and are simply a pair of intersecting lines.
5. Degenerate hyperbolas are a special case of hyperbolas and can be considered a transition between a hyperbola and a pair of intersecting lines.

Review Questions

• Explain the conditions under which a hyperbola becomes degenerate.
  • A hyperbola becomes degenerate when the coefficients of the hyperbola equation are such that the discriminant $b^2 - a^2$ is equal to 0. This means that the two branches of the hyperbola coincide, forming a single straight line. Mathematically, the equation of a degenerate hyperbola can be written in the form $ax^2 + 2bxy + cy^2 = 0$, where $b^2 - ac = 0$. This special case occurs when the hyperbola 'degenerates' into a pair of intersecting lines.
• Describe the key properties of a degenerate hyperbola and how it differs from a standard hyperbola.
  • A degenerate hyperbola is a special case of a hyperbola where the two branches of the hyperbola coincide, forming a single straight line. Unlike a standard hyperbola, which has two separate branches that open in opposite directions, a degenerate hyperbola has no interior points and is simply a pair of intersecting lines. The equation of a degenerate hyperbola can be written in the form $ax^2 + 2bxy + cy^2 = 0$, where $b^2 - ac = 0$. This contrasts with the standard hyperbola equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively.
• Analyze the geometric and algebraic significance of a degenerate hyperbola and explain how it relates to the broader concept of conic sections.
  • From a geometric perspective, a degenerate hyperbola represents a special case where the two branches of the hyperbola coincide, forming a single straight line. Algebraically, this occurs when the coefficients of the hyperbola equation are such that the discriminant $b^2 - a^2$ is equal to 0. The degenerate hyperbola can be seen as a transition between a true hyperbola and a pair of intersecting lines, which are also a type of conic section. Conic sections, in general, are the shapes formed by the intersection of a plane and a cone, and include circles, ellipses, parabolas, and hyperbolas. The degenerate hyperbola, as a special case, highlights the interconnected nature of these fundamental geometric shapes and their underlying algebraic representations.

Related terms

Hyperbola:

A hyperbola is a conic section formed by the intersection of a cone with a plane that cuts through both halves of the cone. It consists of two separate branches that are mirror images of each other and open in opposite directions.

Conic Sections:

Conic sections are the shapes formed by the intersection of a plane and a cone. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas.

Equation of a Hyperbola: The standard equation of a hyperbola in the $x$-$y$ plane is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.