91Ó°ÊÓ

We are given $$x=\cos t$$ $$y=1+\sin t$$ To eliminate the parameter \(t\), we can use the trigonometric identity: \(\sin^2 t + \cos^2 t = 1\). Now we have: $$\cos^2 t = x^2 \Rightarrow \sin^2 t = 1 - x^2$$ From the second equation, we can isolate \(\sin t\) as follows: $$\sin t = y - 1$$ Now, square both sides $$\sin^2 t = (y - 1)^2$$ Substitute the expression for \(\sin^2 t\) into the equation we got from the trigonometric identity, $$1 - x^2 = (y - 1)^2$$

Now we have the equation, $$1 - x^2 = (y - 1)^2$$ We can rearrange it to get the standard form of a circle: $$(y - 1)^2 + x^2 = 1$$

The standard form of a circle is \((x - a)^2 + (y - b)^2 = r^2\), where \((a, b)\) is the center and \(r\) is the radius. Comparing this form with our given equation, $$(y - 1)^2 + x^2 = 1$$ We can see that the center of the circle is \((0, 1)\) and the radius \(r\) is 1.

When \(t\) ranges from \(0 \leq t \leq 2\pi\), the parametric equations describe a circular arc or a full circle. In our case, since \(t\) ranges from 0 to \(2\pi\), we have a complete circle. Now, we need to determine the positive orientation. The positive orientation will be determined by the direction in which the parametric equations progress with increasing \(t\). When we increase \(t\) from 0 to \(\pi / 2\), the \(x\)-coordinate decreases (as \(\cos t\) is decreasing in the first quadrant), and the \(y\)-coordinate increases (because \(\sin t\) is increasing in the first quadrant). So, the circle starts from the point \((1, 1)\) and moves counterclockwise, which tells us that the positive orientation is counterclockwise. Solution:

$$1 - x^2 = (y - 1)^2$$

$$(y - 1)^2 + x^2 = 1$$

Center: \((0, 1)\) Radius: 1

Positive Orientation: Counterclockwise