Question

True or False? Justify your answer with a proof or a counterexample. The equations \(x=\cosh (3 t), y=2 \sinh (3 t)\) represent a hyperbola.

Short answer

True, the equations represent a hyperbola by matching a hyperbola's standard form.

Step-by-step solution

Recall Hyperbola Equation Definition

The standard form of a hyperbola centered at the origin is given by \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). We need to investigate our parametric equations to see if they fit these forms.

Parameterize Using Hyperbolic Identities

Recall that \( \cosh^2(t) - \sinh^2(t) = 1 \) is an identity for hyperbolic functions. Given that \( x = \cosh(3t) \) and \( y = 2 \sinh(3t) \), let's express these parametric equations using this identity.

Express in Terms of Hyperbolic Functions

Substitute the parametric representations: \( x = \cosh(3t) \) and \( y = 2\sinh(3t) \). Thus, \( \cosh(3t) = x \) and \( \sinh(3t) = \frac{y}{2} \).

Apply Hyperbolic Identity

Using the identity \( \cosh^2(3t) - \sinh^2(3t) = 1 \), substitute the values: \( (\cosh(3t))^2 - (\sinh(3t))^2 = 1 \). Translate these into terms of \( x \) and \( y \).

Rewrite the Identity in Terms of x and y

Substitute back: \( (x)^2 - \left(\frac{y}{2}\right)^2 = 1 \). Simplifying gives \( x^2 - \frac{y^2}{4} = 1 \).

Confirm Hyperbola Form

The equation \( x^2 - \frac{y^2}{4} = 1 \) matches the standard form of a hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) with \( a^2 = 1 \) and \( b^2 = 4 \). Therefore, the equations represent a hyperbola.

Key concepts

Parametric Equations

In mathematics, parametric equations are used to express a set of related quantities as explicit functions of an independent variable, known as a parameter. This approach can describe complex geometrical shapes like curves and surfaces.
For hyperbolas, parametric equations provide a way to define the position of points on the curve using hyperbolic functions. In our example, the parametric equations are defined as:

• \( x = \cosh(3t) \)
• \( y = 2 \sinh(3t) \)

Here, the parameter \( t \) can take any real number, allowing the equations to trace out the hyperbola as \( t \) varies. These two expressions directly relate the parameter \( t \) with the hyperbola's coordinates \( (x, y) \). By using parametric equations, we simplify the process of working with curves that may otherwise be hard to describe using standard equations.

Hyperbolic Functions

Hyperbolic functions are analogs of the well-known trigonometric functions but are based on hyperbolas rather than circles. They have similar properties and identities that make them useful in various mathematical areas, including calculus, engineering, and physics.
The two primary hyperbolic functions are the hyperbolic cosine, \( \cosh(t) \), and hyperbolic sine, \( \sinh(t) \). For any real number \( t \), these functions are defined as:

• \( \cosh(t) = \frac{e^t + e^{-t}}{2} \)
• \( \sinh(t) = \frac{e^t - e^{-t}}{2} \)

These functions maintain the important hyperbolic identity which behaves similarly to trigonometric identities and plays a crucial role in understanding hyperbolas. In this context, they help express the coordinates of points on the hyperbola parametrically.

Hyperbolic Identity

One of the most fundamental identities for hyperbolic functions is given by:

• \( \cosh^2(t) - \sinh^2(t) = 1 \)

This identity, reminiscent of the Pythagorean identity for trigonometric functions, is central when working with hyperbolas.
In our problem, we manipulate this identity for the parameter \( 3t \) to explore the relationship between \( x \) and \( y \):

• Substitute: \( x = \cosh(3t) \)
• Substitute: \( y = 2 \sinh(3t) \)

This leads us to

• \( \cosh^2(3t) - (\frac{y}{2})^2 = 1 \)

By using this identity, we connect the parametric equations to the standard form of a hyperbola, confirming the initial hypothesis that they describe a hyperbola.

Standard Form of Hyperbola

The standard form of a hyperbola centered at the origin can take one of two shapes, depending on the axes it opens along:

• Horizontal: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertical: \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \)

In either form, the values \( a^2 \) and \( b^2 \) represent squared measures of the distance from the center to vertices and co-vertices respectively.
In our example, we rewrote the equation using the parametric expressions as:

• \( x^2 - \frac{y^2}{4} = 1 \)

This follows the horizontal standard form, with \( a^2 = 1 \) and \( b^2 = 4 \). Understanding this allows us to correctly identify the shape and orientation of the hyperbola described by the original parametric equations.