Question

the ellipse

$$ \\frac { x ^ { 2 } } { 3 ^ { 2 } } + \\frac { y ^ { 2 } } { 5 ^ { 2 } } = 1 $$

can be drawn counterclockwise with parametric equations. if

$$ x ( t ) = a \\cos ( t ) $$

with a positive, then

a = and y(t) =

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Explanation:

Step1: Recall the standard parametric form of an ellipse

The standard equation of an ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\), and its parametric equations are \(x(t)=a\cos(t)\) and \(y(t)=b\sin(t)\) when drawn counter - clockwise.

Step2: Identify the values of \(a\) and \(b\)

Given the ellipse equation \(\frac{x^{2}}{3^{2}}+\frac{y^{2}}{5^{2}}=1\). Comparing with the standard form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) and the parametric form \(x(t)=a\cos(t)\), we can see that \(a = 3\).
For the \(y\) - component, since the standard parametric form for \(y\) is \(y(t)=b\sin(t)\) and \(b = 5\), so \(y(t)=5\sin(t)\)

Answer:

\(a = 3\) and \(y(t)=5\sin(t)\)