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The problem describes the motion of a particle whose position \((x, y)\) is given by the parametric equations:

\[
x = 1 + \sin(t), \quad y = 2 + 6\cos(t), \quad \text{where} \, \frac{\pi}{2} \leq t \leq 2\pi.
\]

We need to describe the motion of the particle and provide specific points at different stages of the motion. 

### Key Steps to Solve:
1. **Find the starting point at \( t = \frac{\pi}{2} \):**
   \[
   x = 1 + \sin\left(\frac{\pi}{2}\right) = 1 + 1 = 2,
   \]
   \[
   y = 2 + 6\cos\left(\frac{\pi}{2}\right) = 2 + 6(0) = 2.
   \]
   So, the starting point is \( (2, 2) \).

2. **Find the ending point at \( t = 2\pi \):**
   \[
   x = 1 + \sin(2\pi) = 1 + 0 = 1,
   \]
   \[
   y = 2 + 6\cos(2\pi) = 2 + 6(1) = 8.
   \]
   So, the particle moves to the point \( (1, 8) \) at \( t = 2\pi \).

### Motion Description:
The particle moves along an ellipse centered at \( (1, 2) \) since the parametric equations describe an ellipse with:
- Semi-major axis of 6 in the vertical direction (due to the \( 6\cos(t) \)).
- Semi-minor axis of 1 in the horizontal direction (due to the \( \sin(t) \)).

The motion starts at \( (2, 2) \) at \( t = \frac{\pi}{2} \) and moves clockwise three-fourths of the way around the ellipse, ending at \( (1, 8) \).

Would you like further details on the elliptical motion or the parametric equations?

Here are 5 related questions to extend your understanding:
1. What is the general form of parametric equations for an ellipse?
2. How do we determine the direction (clockwise or counterclockwise) of motion for parametric curves?
3. What happens if the parameter range changes from \( t = 0 \) to \( t = 2\pi \) instead?
4. Can we calculate the total length of the particle's path around the ellipse?
5. How can we derive the velocity and acceleration of the particle from the parametric equations?

**Tip:** Parametric equations are a powerful way to represent motion, especially in situations involving curves, such as ellipses and circles.

Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 1 + sin(t), y = 2 + 6 cos(t), pi/2 <= t <= 2pi. The motion of the particle takes place on an ellipse centered at (1, 2). As t goes from pi/2 to 2pi, the particle starts at the point (2, 2) and moves clockwise three-fourths of the way around the ellipse to (1, 8).

This problem involves the motion of a particle in an elliptical path using parametric equations. The particle's position is given as x = 1 + sin(t) and y = 2 + 6 cos(t), with t ranging from pi/2 to 2pi. Starting at point (2, 2) and moving clockwise, it completes three-fourths of the ellipse, ending at (1, 8).

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