Question

For the following equations, determine which of the conic sections is described. $$ x y=4 $$

Short answer

The equation describes a hyperbola.

Step-by-step solution

Identify the General Form

We begin by identifying the general form of the given equation. The equation provided is \( xy = 4 \), which does not directly fit into the standard forms for conics: ellipses, parabolas, hyperbolas, and circles.

Represent the Equation

Rewrite the equation in a form that resembles the conic sections: \( xy = 4 \). Notice that the product of \( x \) and \( y \) is a constant.

Convert to Recognizable Form

We know the equation \( xy = 4 \) can be rewritten in terms of the factored variable form \( xy = c \), which is characteristic of hyperbolas.

Confirm Hyperbola Characteristics

A hyperbola is often described by its characteristic of being able to be expressed as \( xy = k \) (where \( k eq 0 \)). This associates with the presence of two axes of symmetry and a saddle point at the origin.

Key concepts

Hyperbola

Hyperbolas are one of the four types of conic sections, which also include circles, ellipses, and parabolas. They are distinct in appearance and properties. A hyperbola is formed by the intersection of a double cone with a plane in such a way that the angle between the plane and the axis of the cone is less than that made by the plane with the surface of the cone.
Hyperbolas have two separate curves, known as branches. The most apparent feature of a hyperbola is that it has two foci, distinct from each other. The shape of a hyperbola implies an open curve that extends indefinitely in its plane. This is different from ellipses and circles, which are closed curves.
Among its unique features, the hyperbola also has asymptotes. Asymptotes are lines that the curve approaches but never actually meets. These lines give a hyperbola its characteristic shape, ensuring the curve extends infinitely without closing. Hyperbolas can be oriented horizontally or vertically, affecting the position of their branches.

Equation of a hyperbola

The equation of a hyperbola can vary, but it generally falls into a recognizable form. In our exercise, the equation given was \( xy = 4 \), which is a simple form of a hyperbola equation known as a rectangular hyperbola. In this case, it is defined in terms of the product of two variables equaling a constant.

• Standard Form: The standard form of hyperbolas can be \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) for different orientations.
• Rectangular Hyperbola: The form \( xy = k \) represents what is known as a rectangular hyperbola, where the asymptotes are perpendicular.

In this equation, the sign and the coefficients can indicate the orientation and spacing of the hyperbola's branches. It may not always be evident at first glance, but rewriting the equation to fit standard or known forms helps to identify it as a hyperbola. This process of identification is crucial when dealing with equations of conics.

Geometric properties of conics

Conic sections, or conics, are the curves obtained by slicing a three-dimensional double-napped cone with a two-dimensional plane. The resulting shapes are circles, ellipses, parabolas, and hyperbolas. Each of these has distinct geometric properties that define their specific curves and structures.
For hyperbolas:

• They consist of two curves known as branches.
• They have two fixed points called foci. The difference in distances from these foci to any point on the hyperbola remains constant, which is a defining property.
• Hyperbolas also have a center, which is the midpoint of the line segment joining the foci.
• Asymptotes intersect at the center of the hyperbola and determine its shape.

Ellipses and circles, on the other hand, have a closed curve with a constant sum of distances to foci and only one center. Parabolas, with only one focus and vertex, have a unique open and smaller curve compared to hyperbolas. Understanding these properties is essential in identifying and working with various conic sections.