Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{aligned} &x=2 \cosh t \\ &y=2 \sinh t \end{aligned} $$

Short answer

The given parametric equations represent a hyperbola.

Step-by-step solution

Understanding the Parametric Equations

We are given the parametric equations \(x = 2\cosh t\) and \(y = 2\sinh t\). The goal is to determine which type of basic curve these equations describe.

Recall Hyperbolic Functions Identity

Recall the hyperbolic identity \( \cosh^2t - \sinh^2t = 1 \). This identity will help us relate the given parametric equations to a common conic section form.

Substitute and Rearrange

Substitute the parametric equations into the hyperbolic identity. Using \( x = 2 \cosh t \), we get \( \cosh t = \frac{x}{2} \). Similarly, from \( y = 2 \sinh t \), we have \( \sinh t = \frac{y}{2} \).

Setup the Equation

Substitute \( \cosh t = \frac{x}{2} \) and \( \sinh t = \frac{y}{2} \) into the hyperbolic identity: \( \left(\frac{x}{2}\right)^2 - \left(\frac{y}{2}\right)^2 = 1 \).

Simplify and Identify the Curve

Simplify the equation from Step 4 to get \( \frac{x^2}{4} - \frac{y^2}{4} = 1 \), which can be rewritten as \( \frac{x^2}{4} - \frac{y^2}{4} = 1 \). This is the standard form of a hyperbola.

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogous to the trigonometric functions but for a hyperbola instead of a circle. Just as trigonometric functions model circular patterns, hyperbolic functions like \(\cosh\) and \(\sinh\) describe the shape of hyperbolas. You come across two primary functions: hyperbolic cosine \(\cosh\), and hyperbolic sine \(\sinh\). These functions are defined as follows:
\[\cosh t = \frac{e^t + e^{-t}}{2}\]
\[\sinh t = \frac{e^t - e^{-t}}{2}\]
This means if you have \(t\) as a parameter in parametric equations, you can describe movements on a hyperbola using \(x = a \cosh t\) and \(y = b \sinh t\). Where \(a\) and \(b\) are constants that scale the functions.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. You can categorize these curves into four types: circles, ellipses, parabolas, and hyperbolas.

• Circle: All points are equidistant from a center point.
• Ellipse: The sum of distances from two fixed points (foci) to any point on the ellipse is constant.
• Parabola: Any point is equidistant to a fixed point (focus) and a fixed line (directrix).
• Hyperbola: The difference in distances to two fixed points is constant, creating two separate curves.

Each type of conic section can be represented with standard forms of equations, which identify their basic shapes through algebraic expressions.

Hyperbola

Hyperbolas consist of two mirrored arms opening in opposite directions. This distinct shape is defined by the equation
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
The hyperbola's arms reflect around their center at the origin. The constants \(a\) and \(b\) determine the spread and size of these arms. Hyperbolas are unique among conic sections because they consist of two completely separate curves called branches. They diverge as they move away from the origin.

Mathematical Identities

Mathematical identities are equations that hold true for all values within their variables' domain. They are essential for solving and simplifying equations. In the case of hyperbolas, one main identity involves hyperbolic functions:
\[\cosh^2 t - \sinh^2 t = 1\]
This identity can be compared to the Pythagorean identity for trigonometric functions. It is crucial when rewriting parametric equations into standard conic forms. Using identities like these, you can transform and recognize various mathematical forms, such as observing that a set of parametric equations fits the hyperbola equation after applying the identity.