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The task is to find equations for the plane curves described parametrically in parts \(a\) and \(d\) of the problem. Let's analyze and solve each.

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### **Part (a):**
The parametric equation is:
\[
X(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \begin{pmatrix} 2t^2 \\ t \end{pmatrix}, \quad t \in \mathbb{R}.
\]
Here:
- \(x(t) = 2t^2\),
- \(y(t) = t\).

To eliminate \(t\), solve for \(t\) in terms of \(y(t)\):
\[
t = y.
\]
Substitute this into \(x(t)\):
\[
x = 2t^2 = 2y^2.
\]
Thus, the equation of the curve is:
\[
x = 2y^2.
\]

---

### **Part (d):**
The parametric equation is:
\[
X(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \begin{pmatrix} 4\cos(t) \\ 2\sin(t) \end{pmatrix}, \quad t \in \mathbb{R}.
\]
Here:
- \(x(t) = 4\cos(t)\),
- \(y(t) = 2\sin(t)\).

We eliminate \(t\) using the Pythagorean trigonometric identity:
\[
\cos^2(t) + \sin^2(t) = 1.
\]
Rewriting the parametric equations:
\[
\cos(t) = \frac{x}{4}, \quad \sin(t) = \frac{y}{2}.
\]
Substitute these into the identity:
\[
\left(\frac{x}{4}\right)^2 + \left(\frac{y}{2}\right)^2 = 1.
\]
Simplify:
\[
\frac{x^2}{16} + \frac{y^2}{4} = 1.
\]
Thus, the equation of the curve is:
\[
\frac{x^2}{16} + \frac{y^2}{4} = 1,
\]
which represents an ellipse.

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### Final Answer:
1. **(a):** \(x = 2y^2\) (a parabola).
2. **(d):** \(\frac{x^2}{16} + \frac{y^2}{4} = 1\) (an ellipse).

Would you like further clarification or additional examples?

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### Related Questions:
1. What is the general method to eliminate \(t\) in parametric equations?
2. How can you classify curves like parabolas or ellipses using their equations?
3. What are the differences between parametric and Cartesian forms of equations?
4. How would you approach a 3D parametric curve elimination problem?
5. Can any parametric equation be converted into a single Cartesian equation?

### Tip:
For parametric equations, always look for identities (like Pythagorean) or direct substitutions to eliminate the parameter \(t\).

Gib eine Gleichung der ebenen Kurve an, die folgendermaßen beschrieben werden kann: (a) X(t) ∈ ℝ² | X(t) = (2t², t), t ∈ ℝ und (d) X(t) ∈ ℝ² | X(t) = (4cos(t), 2sin(t)), t ∈ ℝ

Discover how to convert parametric equations into Cartesian form for a parabola and ellipse. Solve equations like X(t) = (2t², t) and X(t) = (4cos(t), 2sin(t)) using trigonometric and algebraic techniques. Suitable for Grades 11-12.

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