https://file.aiquickdraw.com/mathbot/file/1730406039915hg90gh4y.jpg

The problem involves analyzing the motion of two particles, each following a specific parametric path over the interval \(0 \leq t \leq 2\pi\).

The given positions are:
1. **Particle 1**: 
   - \( x_1 = 3 \sin(t) \)
   - \( y_1 = 2 \cos(t) \)

2. **Particle 2**:
   - \( x_2 = -3 + \cos(t) \)
   - \( y_2 = 1 + \sin(t) \)

The questions asked are:
1. **(a)** Graphing the paths: The graph options are provided, and it seems the correct answer (based on a check mark) is the third plot.
2. **(b)** Finding intersections and checking for collisions (same position and time):
   - There are two intersection points given: \((-3, 0)\) and \((-2, 1.5)\).
   - It is confirmed that there are no collision points, as none of these points match at the same \( t \)-value for both particles.

3. **(c)** Modifying the second particle's path to:
   - \( x_2 = 3 + \cos(t) \)
   - \( y_2 = 1 + \sin(t) \)
   - Finding the new center of the circular path of the second particle, which is at \((3, 1)\).

Let me know if you need details on each part or have any specific questions.

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### Additional Questions for Practice:
1. How would you determine the parametric equations for the velocity of each particle?
2. What would be the parametric form of the tangent to each path at any given point \( t \)?
3. How can you mathematically determine the exact \( t \)-values at which intersections occur?
4. What changes in the equation for Particle 2 would shift its path vertically by 3 units?
5. Can you determine the speeds of the particles at any point on their paths?

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#### Tip:
When analyzing intersections in parametric equations, ensure both the positions **and** corresponding parameter values (like \( t \)) match to confirm a collision.

Suppose that the position of one particle at time t is given by x1 = 3sin(t), y1 = 2cos(t), 0 ≤ t ≤ 2π and the position of a second particle is given by x2 = -3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2π. (a) Graph the paths of both particles. (b) How many points of intersection are there? Are any of these points collision points? (c) Describe what happens if the path of the second particle is given by x2 = 3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2π.

Learn how to graph parametric paths for two particles and find intersection points. This problem covers advanced concepts in parametric equations, including identifying intersection and collision points. Suitable for Grades 11-12 students.

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