Question on Quadric Surfaces

[Edder]

Hey everyone,
I am having a bit of difficulty understanding a problem. It says: The trace of the graph of z = f(x,y) = x^2 + 2y^2 on the plane z=3, is which of the following conic sections?

At first glance, the equation of z = x^2 + Y^2 is an elliptic paraboloid. However, if I plug in z=3 into the equation, I get x^2 + 2y^2 = 3, which is an ellipse I believe. So, exactly how am I suppose to solve this problem. Am I already correct in assuming the answer is an ellipse? Or is my understanding of the problem incorrect.

Any help and feedback is greatly appreciated, thanks.

 

[tkhunny]

You're not in 2D Kansas any more.

Given a constant z, we have (Constant) = ax^2 + by^2 and we have ellipses in the x-y plane
Given a constant x, we have z = a(Constant)^2 + by^2 and we have parabolas in the y-z plane
Given a constant y, we have z = ax^2 + b(Constant)^2 and we have parabolas in the x-z plane

If we hold nothing cnostant, we have, as you determined, elliptic paraboloid.

 

[Edder]

Thanks, that clears some things up. So basically, depending on the constant that is given (or not given), you can determine the shapes in the xyz plane.

 

[tkhunny]

No such thing. As our human eyes are not always able to relate to the 2D object in the XYZ-plane, we are caused to make these projections in the XY-, XZ-, and YZ-planes. The patern of behavior carries to higher dimensions. A projection to a lower dimensional object aids in understanding.

 

[JeffM]

Thanks, that clears some things up. So basically, depending on the constant that is given (or not given), you can determine the shapes in the xyz plane.

There is no such thing as the x-y-z plane; it is a volume or a space. If you have z = f(x, y), think about the x, y plane being the floor (or ceiling) and the z dimension being above the floor (or below the ceiling). The graph of that function is a surface in space (just as y = g(x) is a curve in the plane).

 

[Edder]

This 3-dimensional space is a bit confusing, but thanks your input.

 

[mmm4444bot]

A projection to a lower dimensional object aids in understanding.

A good analogy of this fact could be the shadow from a moving ceiling fan.

Consider the fan's shadow projected onto a wall from a light source in the room. The shadow is a 2D image.

Now, if a person saw only the shadow on the wall, watching its undulating shape in constant motion, they could likely deduce that the object casting the shadow is a ceiling fan because they recognize things in that two-dimensional shape-shifting that they've seen before in the 3D world.

In other words, they infer information about a 3D object from viewing related 2D information. :cool: