Question

A pair of parametric equations is given. Find a rectangular-coordinate

Equation, expression, and inequalitie
ANSWERED
asked 2021-08-14
A pair of parametric equations is given. Find a rectangular-coordinate equation for the curve by eliminating the parameter.
\(\displaystyle{x}={2}{\cos{{t}}},{y}={3}{\sin{{t}}},{0}≤{t}≤{2}π\)

Expert Answers (1)

2021-08-15

To eliminate the parameter, we use the Pythagorean Identity \(\sin^2 θ + \cos^2 θ = 1\). Solving \(x = 2\cos t\) for \(\cos t\) gives \(\cos t = \frac{x}{2}\). Solving \(y = 3\sin t\) for \(\sin t\) gives \(\sin t = \frac{y}{3}\). Substituting these into the identity then gives:

\((\frac{y}{3})^2 + (\frac{x}{2})^2 = 1\)

\(\frac{y^2}{9} + \frac{x^2}{4} = 1\)

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