Curve of intersection lying in a plane

[MarkSA]

Hello,

Q: Consider the surface:
x^2 + 3y^2 -z^2 + 3x = 1
With which of the following surfaces does the given surface have a curve of intersection that lies in a plane?

I know that two of the correct answers to this are:
2x^2 + 6y^2 - 2z^2 - 9y = 0
and 3x^2 + 9y^2 - 3z^2 - 8y = 0

I'm not at all sure of what procedure to use in this problem. I guess i'm not exactly sure what the question is asking by a 'curve of intersection'. So I tried working this problem out in reverse. Just eyeballing it, it looks like I can take the normal vector for the original surface (plane) to be <1,3,-1>. The 'correct' answers happen to be planes that all appear to have normal vectors that are scalar multiples of the original plane's normal vector. ie: <2,6,-2> and <3,9,-3>

If I remember right this means that all three planes are parallel. But I still don't understand why that makes these answers correct. Can you help me understand this?

 

[mmm4444bot]

Hi Mark:

I regret that I've forgotten too much to work with the equations in this problem; someone else on the board will need to step in.

However, I can explain the phrase "curve of intersection".

Firstly, keep in mind that the word "curve" in calculus describes straight lines as well as curved lines.

Secondly, when two surfaces intersect, the infinite points of intersection form a "seam". This seam is the boundary between the two surfaces. It forms a curve ("line").

Look toward the bottom of the page at the following link; the subheadings "Drop and Cylinder" and "CircleCircle and Simple-Surface" each show the curves of intersection between a blue object and a red object.

I found this by searching the string "intersection surfaces 3-d" at Google

~ Mark

 

[tkhunny]

Multiply the first equation by 2 and subtract from the second equation. What happens?

Multiply the first equation by 3 and subtract from the third equation. What happens?

The intent appears to be that the only modifications allowed are those that preserve relationships orthogonal to the coordinate axes, i.e. expand it, shrink it, and scoot it around, but don't twist it.