Question

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 2 sinh t, y = 2 cosh t, −∞<t<∞

Accepted Answer

Recall the definitions of hyperbolic sine and cosine: $$\sinh t = \frac{e$$^{t}$$ - e$$^{-t}$$}{2}$$ and $$\cosh t = \frac{e$$^{t}$$ + e$$^{-t}$$}{2}. $$These functions satisfy the identity $$\$$cosh^{2}$$ t - \$$sinh^{2}$$ t = 1. $$Given the parametric equations $$x = 2 \sinh t$$ and $$y = 2 \cosh t$$, express $$\sinh t$$ and $$\cosh t in $$terms of $$x$$ and $$y$$: $$\sinh t = \frac{x}{2}$$ and $$\cosh t = \frac{y}{2}. $$Use the hyperbolic identity $$\$$cosh^{2}$$ t - \$$sinh^{2}$$ t = 1$$ and substitute the expressions for $$\sinh t$$ and $$\cosh t to $$get $$\left(\frac{y}{2}\$$right)^{2}$$ - \left(\frac{x}{2}\$$right)^{2}$$ = 1. $$Simplify the equation to obtain the Cartesian form: $$\frac{y$$^{2}$$}{4} - \frac{x$$^{2}$$}{4} = 1$$, which can be rewritten as $$\frac{y$$^{2}$$}{4} - \frac{x$$^{2}$$}{4} = 1 or$$ $$\frac{y$$^{2}$$}{4} - \frac{x$$^{2}$$}{4} = 1. $$Interpret the Cartesian equation as a hyperbola. The parameter $$t$$ ranges over all real numbers, so the particle traces the entire hyperbola. The direction of motion corresponds to increasing $$t$$, which can be analyzed by considering how $$x$$ and $$y$$ change as $$t$$ increases.

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Key Concepts

Parametric Equations

Hyperbolic Functions and Their Identities

Eliminating the Parameter to Find Cartesian Equations