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The problem involves finding the trajectory equation of a moving point, with the parametric equations given as:

\[
\begin{cases}
x = 2t \\
y = t^2 + 1
\end{cases}
\]

To find the trajectory equation \( y \) in terms of \( x \), we can eliminate the parameter \( t \).

### Step 1: Solve for \( t \) in terms of \( x \)
From the equation \( x = 2t \), solve for \( t \):
\[
t = \frac{x}{2}
\]

### Step 2: Substitute \( t \) into the equation for \( y \)
Substitute \( t = \frac{x}{2} \) into the second equation \( y = t^2 + 1 \):
\[
y = \left( \frac{x}{2} \right)^2 + 1
\]
\[
y = \frac{x^2}{4} + 1
\]

### Conclusion:
The equation of the trajectory is:
\[
y = \frac{x^2}{4} + 1
\]
This corresponds to option **b** in the provided choices.

Would you like further details or have any questions?

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Here are 5 related questions to explore:
1. What is the geometric shape of the trajectory based on the equation \( y = \frac{x^2}{4} + 1 \)?
2. How do we derive the general method of eliminating parameters in parametric equations?
3. Can we describe the motion of the particle over time based on these parametric equations?
4. What happens to the equation if the constants are modified (e.g., \( x = 3t \) or \( y = t^2 + 2 \))?
5. How do parametric equations help in representing complex motions in physics?

**Tip**: When working with parametric equations, solving for one variable in terms of the parameter can simplify the process of finding the overall trajectory.

Một chất điểm có phương trình chuyển động theo các trục như sau: x = 2t, y = t^2 + 1. Phương trình quỹ đạo của chất điểm có dạng là?

Learn how to find the trajectory equation from the parametric equations x = 2t and y = t^2 + 1. This problem involves key algebraic concepts and the method of eliminating the parameter. Suitable for students in Grades 10-12.

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