Explanation:

The given curve is represented by the parametric equations:

x = 3(cos t)(sin t)
y = 4(cos t - sin t)

To determine the shape of the curve, we need to analyze the equations and identify the type of conic section it represents.

1. Equation Analysis:

Let's simplify the equations to gain a better understanding:

x = 3(cos t)(sin t) = 3/2 * sin 2t
y = 4(cos t - sin t) = 4 * cos (t + π/4)

2. Elimination of Parameter:

To eliminate the parameter t, we can square both equations and add them together:

x² = (3/2)² * sin² 2t
y² = 4² * cos² (t + π/4)

Adding these equations:

x² + y² = (9/4) * sin² 2t + 16 * cos² (t + π/4)

3. Trigonometric Identity:

Using the identity sin² θ + cos² θ = 1, we can rewrite the equation as:

x² + y² = (9/4) * (1 - cos² 2t) + 16 * (1 - sin² (t + π/4))

Simplifying further:

x² + y² = (9/4) - (9/4) * cos² 2t + 16 - 16 * sin² (t + π/4)

x² + y² = (25/4) - (9/4) * cos² 2t - 16 * sin² (t + π/4)

4. Simplification:

By applying trigonometric identities, we can simplify the equation even further:

x² + y² = (25/4) - (9/4) * (1 - 2sin² 2t) - 16 * (1 - cos² (t + π/4))

x² + y² = (25/4) - (9/4) + (18/4) * sin² 2t - 16 + 16 * cos² (t + π/4)

x² + y² = (18/4) * sin² 2t + 16 * cos² (t + π/4)

x² + y² = 9 * sin² 2t + 16 * cos² (t + π/4)

5. Conic Section Identification:

Comparing the equation with the standard forms of conic sections, we observe:

x² + y² = A² * sin² θ + B² * cos² θ

This equation represents an ellipse, where A and B are positive constants.

Conclusion:

Therefore, the curve represented by the given parametric equations is an ellipse.

This curve represents ellipse