# Solutions to Problem Set 11

Math 311-01/02

2-12-2018

[Parametrizing curves problems]

1. Parametrize the segment from the point to the point .

2. Parametrize the circle

3. Parametrize the ellipse

Here's the idea: Divide by 16 and write the terms on the left as squares:

If I "match" the last equation against the identity , I get

Solving for x and y gives the parametrization:

4. Parametrize the curve of intersection of the cylinder with the plane .

Thinking of as a curve in the x-y plane, I may parametrize it by

For the plane, , so

The curve is

5. Parametrize the curve of intersection of the elliptical cylinder with the surface .

Thinking of as a curve in the x-y plane, I may parametrize it by

Plugging this into gives

The curve is

6. Parametrize the curve of intersection of the cylinder with the hemisphere .

First, complete the square:

This is a circle of radius 1 with center .

Thinking of this as a curve in the x-y plane, I may parametrize it by

Plugging this into gives

The curve is

7. Parametrize the curve of intersection of the x-z plane with the surface .

The intersection of the surfaces is obtained by setting in . This gives . Set , so . The curve is

8. Parametrize the curve of intersection of the surface with the surface .

Thinking of as a curve in the x-y plane, I may parametrize it by

Plugging this into , I get

The curve is

Hope is a very unruly emotion. - Gloria Steinem

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