Which of the following is (are) a pair of parametric equation that represent a circle?
`{:{(x=sin theta),(y=cos theta):}`
`{:{(x=t),(y=sqrt(1-t^(2))):}`
`{:{(x=sqrt(s)),(y=sqrt(1-s)):}`

Options

A. only I
B. only II
C. only III
D. only II and III

Explanation

To determine which of the given pairs of parametric equations represent a circle, we need to check each pair against the standard equation of a circle, which is given by:

\[ x^2 + y^2 = r^2 \]

where \( r \) is the radius of the circle.

### Step-by-Step Solution:

1. **First Pair: \( x = \sin \theta, y = \cos \theta \)**

Substitute \( x \) and \( y \) into the circle equation:
\[ x^2 + y^2 = \sin^2 \theta + \cos^2 \theta \]
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we have:
\[ x^2 + y^2 = 1 \]
This represents a circle of radius 1. Therefore, the first pair of parametric equations does represent a circle.

2. **Second Pair: \( x = t, y = \sqrt{1 - t^2} \)**

Substitute \( x \) and \( y \) into the circle equation:
\[ x^2 + y^2 = t^2 + (\sqrt{1 - t^2})^2 \]
Simplifying this gives:
\[ t^2 + (1 - t^2) = 1 \]
Therefore:
\[ x^2 + y^2 = 1 \]
However, since \( y = \sqrt{1 - t^2} \), \( y \) is always non-negative. This means that the graph only represents the upper half of the circle (a semicircle) and not the full circle. Thus, the second pair does not represent a complete circle.

3. **Third Pair: \( x = \sqrt{s}, y = \sqrt{1 - s} \)**

Substitute \( x \) and \( y \) into the circle equation:
\[ x^2 + y^2 = (\sqrt{s})^2 + (\sqrt{1 - s})^2 \]
This simplifies to:
\[ s + (1 - s) = 1 \]
Thus:
\[ x^2 + y^2 = 1 \]
Similar to the second pair, both \( x \) and \( y \) are non-negative (since they are square roots), which means this also represents only the upper half of the circle. Therefore, the third pair does not represent a complete circle.

### Conclusion:

- The first pair of parametric equations \( (x = \sin \theta, y = \cos \theta) \) represents a circle.
- The second pair \( (x = t, y = \sqrt{1 - t^2}) \) represents a semicircle.
- The third pair \( (x = \sqrt{s}, y = \sqrt{1 - s}) \) also represents a semicircle.

**Final Answer: Only the first pair represents a circle.**