MHB Are the Intersections of These 3D Curves Ellipses?

[carl123]

QUESTION 1

Sketch the curves :

z = x² + y² and z = 2x² + 3y² - 1

a) Is the intersection an ellipse?

b) Is the projection of the intersection onto the x-y plane an ellipse?

QUESTION 2

Sketch the curves:

r1[t] = (t cos [2 pi t], t sin [2 pi t] , t2 ) and r2[t] = (t cos [6 pi t], t sin [6 pi t], t2 ] over the intervals from 0 to 2.

a) There is a quadratic surface that both curves lie on, what sort of surface is it? Give the name?

b) What is an equation for the surface?

 

[Prove It]

QUESTION 1

Sketch the curves :

z = x² + y² and z = 2x² + 3y² - 1

a) Is the intersection an ellipse?

b) Is the projection of the intersection onto the x-y plane an ellipse?

QUESTION 2

Sketch the curves:

r1[t] = (t cos [2 pi t], t sin [2 pi t] , t2 ) and r2[t] = (t cos [6 pi t], t sin [6 pi t], t2 ] over the intervals from 0 to 2.

a) There is a quadratic surface that both curves lie on, what sort of surface is it? Give the name?

b) What is an equation for the surface?

Have you tried anything?

To sketch $\displaystyle \begin{align*} z = x^2 + y^2 \end{align*}$, think about every cross section parallel to the x-y plane. First of all, for obvious reasons, $\displaystyle \begin{align*} z \geq 0 \end{align*}$. Also obviously no matter what z value you pick, you are going to have something of the form $\displaystyle \begin{align*} x^2 + y^2 = r^2 \end{align*}$ (r a constant), therefore each cross section must be a circle. But since as you move up the z axis, you are increasing z, that means r is increasing as well. So each cross section is a circle, getting bigger in radius as you move up the z axis. That seems like a paraboloid to me...

z '=' x'^'2 '+' y'^'2 - Wolfram|Alpha

What do you think the second surface would look like and be classified as?

As for finding their intersection, well you already have both of them as z in terms of x and y, so you can set them equal to each other...