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. Start with the given parametric equations of the hyperbola: \(x = h + a \sec t\) and \(y = k + b \tan t\).
2. Recall the fundamental trigonometric identity relating secant and tangent: \(\sec^2 t - \tan^2 t = 1\).
3. Express \(\sec t\) and \(\tan t\) in terms of \(x\) and \(y\) by isolating them from the parametric equations: \(\sec t = \frac{x - h}{a}\) and \(\tan t = \frac{y - k}{b}\).
4. Substitute these expressions into the identity \(\sec^2 t - \tan^2 t = 1\) to eliminate the parameter \(t\).
5. Simplify the resulting equation to write it in the standard form of a hyperbola: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how to manipulate these equations is essential for eliminating the parameter and finding a direct relationship between x and y.

Trigonometric Identities Involving Secant and Tangent

The identity sec²t - tan²t = 1 is crucial when working with parametric equations involving sec t and tan t. This identity allows the elimination of the parameter t by relating sec t and tan t algebraically.

Standard Form of a Hyperbola

The standard form of a hyperbola centered at (h, k) is ((x - h)² / a²) - ((y - k)² / b²) = 1. Recognizing this form helps in rewriting the equation after eliminating the parameter to identify the conic section clearly.