Problem 57 - Calculus Single Variable

Question

Identify the curve that is parameterized by the equations. In each case, use a trigonometric identity to eliminate the parameter. (It may help to consider the special case \(a=b=1\) first.) \(x=a \tan (\theta), y=b \cot (\theta), \theta \in(0, \pi / 2)\)

Step-by-Step Solution

Answer

The curve is a hyperbola given by the equation \(xy = ab\).

Step 1: Understand the Parametric Equations

The parametric equations are given as \(x = a \tan(\theta)\) and \(y = b \cot(\theta)\). We need to find a relationship between \(x\) and \(y\) that does not involve the parameter \(\theta\).

Step 2: Use Trigonometric Identities

Recall that the identity for cotangent is \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Thus, \(y = b \frac{1}{\tan(\theta)} = b\frac{1}{x/a}\) when expressed in terms of \(x\).

Step 3: Substitute and Simplify

Substitute \(x = a \tan(\theta)\) into the equation for \(y\) to get \(y = \frac{b}{x/a}\). Simplifying this equation, we get \(xy = ab\).

Step 4: Characterize the Curve

From the equation \(xy = ab\), we see that this is an equation of a hyperbola, as it expresses a constant product of \(x\) and \(y\).

Key Concepts

Trigonometric Identities

A crucial step in solving parametric equations involves understanding and using trigonometric identities. In this exercise, the key identity used is related to the tangent and cotangent functions. Recall that the cotangent is simply the reciprocal of tangent:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Using this identity helps us express one parameterized equation in terms of another by eliminating the parameter \( \theta \).

This transformation is pivotal when you aim to link seemingly disparate equations into a single, cohesive one. By recognizing these identities, you can manipulate the equations to reveal underlying geometric shapes or mathematical properties. In this case, it clarified the relationship between \( x \) and \( y \), making it easier to identify the curve type.

Hyperbola

The relationship \( xy = ab \) unveils the nature of the curve—a hyperbola. Hyperbolas are conic sections characterized by the difference of distances from any point on the hyperbola to two fixed points (foci) being a constant.
A typical equation for a hyperbola looks like this:

• Standard form: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Parametrization focuses on another form: \( xy = c \) (as in this scenario)

The expression \( xy = ab \) is structurally similar to this simplified version, indicating a rectangular hyperbola when \( a = b \). These relationships between elements \( x \) and \( y \) highlight the spatial symmetry and geometry intrinsic to hyperbolas.

Parameter Elimination

Parameter elimination is a technique used to convert parametric equations into a standard form that doesn't involve the parameter (in this case, \( \theta \)). This approach allows us to study the curve's properties in a more straightforward manner.

• It starts by expressing one variable explicitly in terms of the parameter.
• Then, it substitutes this expression into the equation of the other variable.
• Finally, it simplifies the result to highlight the geometric relationship between variables.

In the original exercise, replacing \( \tan(\theta) \) with \( x/a \) and subsequently adjusting \( y \) allows for the elimination of \( \theta \). This transforms the equations into \( xy = ab \), hence revealing the curve's nature as a hyperbola.
Mastering parameter elimination is essential for simplifying and understanding the geometry of parametrically defined curves.