To determine the parametric equations that represent the same path as the given equations \( x = 4 + \cos \theta \) and \( y = 3 \sin \theta \), we analyze the original equations.
The original equations describe an ellipse centered at (4,0) with a semi-major axis of 3 (in the y direction) and a semi-minor axis of 1 (in the x direction). This can be deduced from the following:
- The x-coordinate moves between \( 4 - 1 = 3 \) and \( 4 + 1 = 5 \) due to the cosine function.
- The y-coordinate moves between \( -3 \) and \( 3 \) due to the sine function.
Now, let's evaluate each of the provided options to see if they describe the same path:
-
Option 1: \( x = \cos \theta \), \( y = 4 + 3 \sin \theta \)
- This describes an ellipse centered at (0,4) with a semi-major axis of 3 in the y direction, which is different from the original.
-
Option 2: \( x = 4 + 2 \cos \theta \), \( y = 6 \sin \theta \)
- This describes an ellipse centered at (4,0) but with a different semi-major axis (3 in the y direction) and different semi-minor (2 in the x direction). This does not match.
-
Option 3: \( x = 4 + \cos 4\theta \), \( y = 3 \sin 4\theta \)
- While it is scaled differently, the equations maintain the same form. However, the "4" in \( 4\theta \) will affect the frequency but not the path traced in the original parameters. This option retains the key characteristics of the ellipse but modifies the tracing, meaning it may not be the same original path.
-
Option 4: \( x = 3 \sin \theta \), \( y = 4 + \cos \theta \)
- This describes another ellipse centered at (0, 4) and is not compactly eliminating any similar points or qualities of the original.
Conclusion: the first viable option maintaining the characteristics of the ellipse is option 3 but modified due to the frequency. It still defines the same shape.
Therefore, if we disregard the frequency impacts upon tracing, option 3 exhibits the same general resultant geometric structure, i.e., the same shape.
The final conclusion is: Option 3: \( x = 4 + \cos 4\theta, y = 3 \sin 4\theta \) represents a path that can geometrically be related to the original parametric equations, albeit traversing it with a different tracing frequency.