The graph represented by the equations ` x= sin ^(2) t, y = 2 cos t ` is

Options

A. a part of parabola
B. a parabola
C. a part of sine graph
D. pan of hyperbola

Answer

To find the graph represented by the parametric equations \( x = \sin^2 t \) and \( y = 2 \cos t \), we can eliminate the parameter \( t \) and express the relationship between \( x \) and \( y \).

### Step-by-Step Solution:

1. **Identify the parametric equations**:
\[ x = \sin^2 t \]
\[ y = 2 \cos t \]

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

3. **Use the Pythagorean identity**:
We know from trigonometry that:
\[ \sin^2 t + \cos^2 t = 1 \]
Substitute \( \cos t \) from step 2 into this identity:
\[ \sin^2 t + \left(\frac{y}{2}\right)^2 = 1 \]

4. **Substitute \( \sin^2 t \) with \( x \)**:
Since \( x = \sin^2 t \), we can replace \( \sin^2 t \) in the equation:
\[ x + \left(\frac{y}{2}\right)^2 = 1 \]

5. **Rearrange the equation**:
This can be rearranged to:
\[ x + \frac{y^2}{4} = 1 \]
or
\[ \frac{y^2}{4} = 1 - x \]

6. **Multiply through by 4**:
To express it in a standard form, multiply the entire equation by 4:
\[ y^2 = 4(1 - x) \]

7. **Final form**:
The equation can be rewritten as:
\[ y^2 = -4x + 4 \]
This is the equation of a parabola that opens to the left.

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