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

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Start with the given parametric equations of the hyperbola: \(x = h + a \sec t\) and \(y = k + b \tan t\).

Recall the fundamental trigonometric identity relating secant and tangent: \(\sec^2 t - \tan^2 t = 1\).

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 \(\tan t = \frac{y - k}{b}\).

Substitute these expressions into the identity \(\sec^2 t - \tan^2 t = 1\) to eliminate the parameter \(t\).

Simplify the resulting equation to write it in the standard form of a hyperbola: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

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주요 개념

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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.

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