Question

c6-parametric equations: problem 3
(1 point)
write the parametric equations

$x = 4 e ^ { t }, quad y = 4 e ^ { - t }$

as a function of $x$ in cartesian form.

$y =$ with $x > 0$
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Explanation:

Step1: Solve \(x = 4e^{t}\) for \(e^{t}\)

From \(x = 4e^{t}\), we get \(e^{t}=\frac{x}{4}\).

Step2: Use the property of exponents

Since \(y = 4e^{-t}\) and \(e^{-t}=\frac{1}{e^{t}}\), substitute \(e^{t}=\frac{x}{4}\) into the expression for \(y\). Then \(y = 4\times\frac{1}{e^{t}}\).

Step3: Substitute \(e^{t}\) value

Substitute \(e^{t}=\frac{x}{4}\) into \(y = 4\times\frac{1}{e^{t}}\), we have \(y = 4\times\frac{4}{x}\).

Answer:

\(y=\frac{16}{x}\)