What kind of geometric object does $3x^2+4y^2-12x+8y+17z=0$ represent?

Question

I am given the following equation $$3x^2+4y^2-12x+8y+17z=0$$ and I have to find what kind of geometric object this is.

I don't think this is a sphere because I can't find the radius or the center from the given information. And I also don't think that this is a cylindrical surface because it contains all the variables $x$, $y$, and $z$. But I may be wrong.

Could you please explain what this is? Thank you very much

Answer 1

I believe it's an elliptic parabloid.

I think if you complete the square you will get something more decipherable like this:

$$3(x-2)^2 +4(y+1)^2 +17z -16 = 0$$

Answer 2

It is elliptical paraboloid:

• For each fixed $z\leq {16\over 17}$ you get an ellipse:

$$3(x-2)^2 + 4(y+1)^2 = 16-17z$$

• For each fixed $x$ or $y$ you get a parabola.

Answer 3

It is an elliptic paraboloid.

If you know the canonical equation of an elliptic paraboloid, you can show this by completing the square.

Answer 4

Mathematica image

p = 7 ; ContourPlot3D[ 3 x^2 + 4 y^2 - 12 x + 8 y + 17 z = = 0, {x, -p, p}, {y, -p, p}, {z, -p, p}]

Could be reducible to form

$$\dfrac{z}{c} = 1 -\dfrac{( x-p)^2}{a^2}-\dfrac{ (y-q)^2}{b^2} ...$$

Elliptic paraboloid as the sections parallel to axes are ellipses, parabolas..