Parameterization of a curve (line & circle)

Question

Let the curve C be the boundary of the region bounded by $y=2$ and $x^2+y^2=9$.

a) How would C look like?

(I think it would be circle segment?)

b) Which of the following characteristics does C have?

• smooth

• piece wise smooth

• Jordan curve

• closed

(I think it is closed and smooth and also a Jordan curve?) But how can I prove this mathematically?

c) I need to find a parameterization of the curve, and here I do not know how to start, do I parameterize the line and the circle arc separately?

the intersection points would be $P(\sqrt{5} | 2), Q(-\sqrt{5} | 2)$

Answer 1

For the parametrization, if it is the upper sliver, you can solve the quadratic equation for $y$:

$y=\pm\sqrt{9-x^2}$ and since we’re working in $y>0$, simply $y=\sqrt{9-x^2}.$

Then letting $x=t$ we have the parametrization $(t,\sqrt{9-t^2})$ for $-\sqrt{5}\leq t\leq\sqrt{5}$.

Your line is $y=2$ thus the parametrization $(t,2)$ for $-\sqrt{5}\leq t\leq\sqrt{5}$.

These are easy ways to think of parameterizations at first, you can work your way to different methods such as polar coordinates.

Answer 2

Rather than giving an explicit answer, I'll give you a plan on as how to think of this yourself.
The first thing I would recommend is drawing the two boundaries, you will earn lots of intuition about the problem.
Once you've done that, you will see the curve is a circle segment AND the segment joining the two ends since the radius is 3 > 2. Therefore it's not smooth, but piecewise-smooth since it fails on those two points (you will see it graphically), it is closed and a Jordan Curve.
As for the parametrization, draw the boundaries, and parametrize those two individually and then join them continuously. With that parametrization you can actually prove (b)