Solving Quadric Surfaces: Reducing, Classifying & Sketching

[DWill]

Homework Statement

Reduce the equation to one of the standard forms, classify the surface, and sketch it:

4x = y^2 - 2z^2

Homework Equations

The Attempt at a Solution

I really don't know what to do for this one because most of the equations I've seen like this involved x^2.

Unrelated to this question: For doing these kinds of problems do you find the cross sections on each plane and then sketch it? For example, if the equation is x^2 + 4y^2 + z^2 = 4, you set one variable at a time to k:

x=k: 4y^2 + z^2 = 4 - k^2
y=k: x^2 + z^2 = 4 - 4k^2
z=k: x^2 + 4y^2 = 4 - k^2

So you can see the cross sections on each plane will be ellipses?

--------------------------------------

2nd problem:

Homework Statement

Find an equation for the surface obtained by rotating the line x = 3y about the x-axis.

The Attempt at a Solution

I know this is a cone about the x-axis, but not sure how to get the exact equation.

 

[D H]

I really don't know what to do for this one because most of the equations I've seen like this involved x^2.

So change the names of the variables.

Look at it this way: The standard form for a parabola is y=ax^2+bx+c. x=ay^2+by+c is also a parabola.