The parametric representation `(2+t^2,2t+1)`represents

Options

A. a parabola with focus at `(2, 1)`
B. a parabola with vertex at `(2,1)`
C. an ellipse with center at `(2,1)`
D. none of these

Answer

To solve the problem, we need to analyze the given parametric representation of the curve, which is \((2 + t^2, 2t + 1)\).

### Step-by-Step Solution:

1. **Identify the Parametric Equations**:
Let \( x = 2 + t^2 \) and \( y = 2t + 1 \).

2. **Express \( t \) in terms of \( y \)**:
From the equation for \( y \):
\[ y = 2t + 1 \implies 2t = y - 1 \implies t = \frac{y - 1}{2} \]

3. **Substitute \( t \) into the equation for \( x \)**:
Substitute \( t = \frac{y - 1}{2} \) into the equation for \( x \):
\[ x = 2 + t^2 = 2 + \left(\frac{y - 1}{2}\right)^2 \]
Simplifying this gives:
\[ x = 2 + \frac{(y - 1)^2}{4} \]

4. **Rearranging the equation**:
Rearranging the equation to express it in standard form:
\[ x - 2 = \frac{(y - 1)^2}{4} \]
Multiplying both sides by 4:
\[ 4(x - 2) = (y - 1)^2 \]

5. **Identify the form of the equation**:
The equation \( (y - 1)^2 = 4(x - 2) \) is in the standard form of a parabola:
\[ (y - k)^2 = 4a(x - h) \]
where \( (h, k) \) is the vertex of the parabola.

6. **Determine the vertex**:
From the equation, we can identify:
- \( h = 2 \)
- \( k = 1 \)

Thus, the vertex of the parabola is \( (2, 1) \).

### Conclusion:
The parametric representation \( (2 + t^2, 2t + 1) \) represents a parabola with vertex at \( (2, 1) \).