To eliminate the parameter $t$ from the parametric equations $x(t) = 3\sin(4t)$ and $y(t) = 5\cos(4t)$, we use the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$.

Step 1: Solve for $\sin(4t)$ and $\cos(4t)$ in terms of $x$ and $y$

From $x(t) = 3\sin(4t)$: $\sin(4t) = \frac{x}{3}.$

From $y(t) = 5\cos(4t)$: $\cos(4t) = \frac{y}{5}.$

Step 2: Apply the Pythagorean identity

$\sin^2(4t) + \cos^2(4t) = 1.$

Substitute $\sin(4t) = \frac{x}{3}$ and $\cos(4t) = \frac{y}{5}$: $\left(\frac{x}{3}\right)^2 + \left(\frac{y}{5}\right)^2 = 1.$

Step 3: Simplify the equation

$\frac{x^2}{9} + \frac{y^2}{25} = 1.$

This is the equation of an ellipse with:

• Semi-major axis $a = 5$ along the $y$-axis.
• Semi-minor axis $b = 3$ along the $x$-axis.

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Step 4: Graph the ellipse

Let me plot the ellipse based on the equation $\frac{x^2}{9} + \frac{y^2}{25} = 1$. numpy as np import matplotlib.pyplot as plt

Define the parameters for the ellipse

a = 5 # Semi-major axis (y-direction) b = 3 # Semi-minor axis (x-direction)

Parametric angle

t = np.linspace(0, 2 * np.pi, 500)

Ellipse points

x = b * np.cos(t) y = a * np.sin(t)

Plot the ellipse

plt.figure(figsize=(8, 8)) plt.plot(x, y, label=r'$\frac{x^2}{9} + \frac{y^2}{25} = 1$', color='blue') plt.axhline(0, color='black',linewidth=0.5, linestyle='--') plt.axvline(0, color='black',linewidth=0.5, linestyle='--') plt.gca().set_aspect('equal', adjustable='box') plt.title('Graph of the Ellipse', fontsize=14) plt.xlabel('x', fontsize=12) plt.ylabel('y', fontsize=12) plt.legend(fontsize=12) plt.grid(True) plt.show()Here is the graph of the ellipse defined by the equation $\frac{x^2}{9} + \frac{y^2}{25} = 1$. The semi-major axis extends along the $y$-axis, and the semi-minor axis extends along the $x$-axis.

Would you like further details on any aspect of this solution or its derivation?

Here are 5 related questions for further exploration:

1. What are the foci of this ellipse, and how can they be calculated?
2. How do we parameterize a rotated ellipse in terms of $t$?
3. What is the area enclosed by this ellipse?
4. How does the equation of the ellipse change if it is translated from the origin?
5. Can this ellipse be described in polar coordinates? How?

Tip: For any ellipse, the sum of the distances from any point on the ellipse to the two foci is always constant.