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To find the center of the circle traced by the particle, we can examine the parametric equations given:

\[
x_2 = 3 + \cos(t)
\]
\[
y_2 = 1 + \sin(t)
\]

These equations describe the \( x \)- and \( y \)-coordinates of the second particle in terms of \( t \), where \( t \) varies from \( 0 \) to \( 2\pi \).

The terms \( \cos(t) \) and \( \sin(t) \) suggest that the particle follows a circular path since they represent a circular motion pattern. The parameters \( 3 \) and \( 1 \) added to \( \cos(t) \) and \( \sin(t) \), respectively, indicate the center of this circular motion.

1. **Center of the Circle**:
   - The circle is centered at \( (3, 1) \).

2. **Radius of the Circle**:
   - Since the path equations \( \cos(t) \) and \( \sin(t) \) have coefficients of \( 1 \), the radius of the circle is \( 1 \).

Thus, the circle traced by the second particle has:
- **Center**: \( (3, 1) \)
- **Radius**: \( 1 \)

Would you like further details on how these parameters affect the motion?

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Here are some related questions to deepen understanding:

1. What would happen to the path if we changed the center to \( (4, 2) \)?
2. How does changing the radius impact the particle's path?
3. What if \( \cos(t) \) and \( \sin(t) \) were scaled by a factor other than 1?
4. Could the path be an ellipse if different coefficients were applied to \( \cos(t) \) and \( \sin(t) \)?
5. How would you calculate the velocity of the particle along this path?

**Tip:** Shifting the center in parametric equations allows you to translate the path without changing its shape.

Explore how to find the center and radius of a circle given parametric equations x = 3 + cos(t) and y = 1 + sin(t). This problem involves parametric equations and basic trigonometry, ideal for students in Grades 10-12.

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