QUADRIC SURFACES

[henrry]

URGENT HELP FOR QUADRIC SURFACES
Identify the surface whose equation in rectangular coordinates is given by:
Z=x^2 +y^2-4x+2y+6
Show its vertex, axis of symmetry, traces in the xy, xz, and yz-planes and name
Thank you for your prompt help at your earliest convenience.

 

[tkhunny]

Have you considered completing the square in x and then in y?

Traces are easy. These should not be stumping you.

For the xz-plane, simply set y = 0 and you should recognize the planer figure.
For the yz-plane, simply...you're not goiong to make me say it all, are you?

No kidding. You have to be able to do these simple traces - ones that you should have studied extensively in your analytic geometry (pre calculus) days. If you can't just pop these our, you will not be a happy camper when they get less familiar.

 

[BigGlenntheHeavy]

\(\displaystyle Note: \ When \ dealing \ with \ Quadric \ Surfaces, \ it \ behooves \ one \ to \ apply \ a \ little \ ingenuity.\)

\(\displaystyle Now, \ z \ = \ x^{2}-4x+y^{2}+2y+6 \ = \ (x-2)^{2}+(y+1)^{2}+6-4-1 \ \implies \ z-1 \ = \ (x-2)^{2}+(y+1)^{2}\)

\(\displaystyle Hence, \ we \ have \ a \ circle \ on \ the \ xy \ plane \ with \ r^{2} \ = \ z-1 \ \implies \ z \ \ge1\)

\(\displaystyle Ergo, \ when \ z \ =1 \ we \ have \ a \ point \ at \ (2,-1,1) \ emanating \ upward \ into \ a \ circle,\)

\(\displaystyle This \ tells \ us \ that \ we \ are \ dealing \ with \ a \ circular \ paraboloid \ with \ vertex \ (2,-1,1).\)

\(\displaystyle Now, \ can \ you \ take \ it \ from \ here?\)

Implicit plot of f(x,y)

[attachment=0:2kmkq954]abc.jpg[/attachment:2kmkq954]