The graph represented by `x=sin^2t, y=2cost` is

Options

A. a protion of a parabola
B. a part of a hyperbola
C. a part of a sing graph
D. a part of a circle

Answer

To solve the problem, we need to analyze the given parametric equations \( x = \sin^2 t \) and \( y = 2 \cos t \). We will eliminate the parameter \( t \) to find the relationship between \( x \) and \( y \).

### Step-by-Step Solution:

1. **Express \( \sin^2 t \) and \( \cos t \)**:
- We have \( x = \sin^2 t \).
- From this, we can express \( \sin t \) as \( \sin t = \sqrt{x} \) (since \( \sin^2 t \) is non-negative).
- We also know that \( y = 2 \cos t \).

2. **Use the Pythagorean Identity**:
- We know from trigonometry that \( \sin^2 t + \cos^2 t = 1 \).
- We can express \( \cos^2 t \) in terms of \( x \):
\[ \cos^2 t = 1 - \sin^2 t = 1 - x \]

3. **Express \( \cos t \)**:
- Taking the square root gives us \( \cos t = \sqrt{1 - x} \) or \( \cos t = -\sqrt{1 - x} \).
- Since \( y = 2 \cos t \), we can substitute:
\[ y = 2\sqrt{1 - x} \quad \text{or} \quad y = -2\sqrt{1 - x} \]

4. **Square the Equation**:
- To eliminate the square root, we can square both sides:
\[ \left(\frac{y}{2}\right)^2 = 1 - x \]
- This simplifies to:
\[ \frac{y^2}{4} = 1 - x \]

5. **Rearranging the Equation**:
- Rearranging gives us:
\[ x + \frac{y^2}{4} = 1 \]
- This can be rewritten as:
\[ \frac{y^2}{4} = 1 - x \]
- Or:
\[ y^2 = 4(1 - x) \]

6. **Identify the Type of Graph**:
- The equation \( y^2 = 4(1 - x) \) represents a parabola that opens to the left with its vertex at the point \( (1, 0) \).

### Conclusion:
The graph represented by the equations \( x = \sin^2 t \) and \( y = 2 \cos t \) is a portion of a parabola.