In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.


A hyperbola: x = h + a sec t, y = k + b tan t

Verified step by step guidance

1. Identify the parametric equations: \( x = h + a \sec t \) and \( y = k + b \tan t \).
2. Express \( \sec t \) in terms of \( x \): \( \sec t = \frac{x - h}{a} \).
3. Express \( \tan t \) in terms of \( y \): \( \tan t = \frac{y - k}{b} \).
4. Use the identity \( \sec^2 t - \tan^2 t = 1 \) to eliminate the parameter \( t \).
5. Substitute \( \sec t \) and \( \tan t \) into the identity and simplify to obtain the equation of the hyperbola in standard form.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, often denoted as 't'. In this case, x and y are defined in terms of the parameter 't', which allows for the representation of complex shapes like hyperbolas. Understanding how to manipulate these equations is crucial for eliminating the parameter and finding a relationship between x and y.

Hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are defined by their standard form equations. Recognizing the characteristics of hyperbolas, such as their asymptotes and the relationship between their axes, is essential for converting parametric equations into standard form.

Standard Form of a Conic Section

The standard form of a conic section provides a concise way to express the geometric properties of the shape. For hyperbolas, the standard form is typically written as (x-h)²/a² - (y-k)²/b² = 1, where (h, k) is the center, and 'a' and 'b' are the distances to the vertices and co-vertices, respectively. Converting parametric equations to this form allows for easier analysis and graphing of the hyperbola.