Curve of Intersection: Finding the Intersection Between a Sphere and an Ellipse

[kamino]

Homework Statement

I am asked to find the curve of intersection between x^2 + y^2 + z^2 = 36 and 2x + y -z = 2.

Homework Equations

The Attempt at a Solution

I know the first equation is a sphere of radius 6, and the second equation will pass through it, so when projected on the xy plane it will also be an ellipse. I'm just not sure how to go about solving for the (x,y) coordinates, so that I know the curve of the line. My guess would be to solve for z in both equations and set them equal to each other. However, when I do that I end up with something ugly...

36 = 5x^2 + 2y^2 + 4xy -8x -4y + 4

Then I'm stuck...

 

[dynamicsolo]

36 = 5x^2 + 2y^2 + 4xy -8x -4y + 4

I get this also. You now have the equation for an ellipse, the axes of which are rotated with respect to the x- and y-axes. (That's what the mixed term, Bxy, tells us.) If that's all you need to know, you're done.

A note on the test for identifying the type of conic section is found at the end of this entry: http://en.wikipedia.org/wiki/Rotation_of_axes . There is also a discussion of what you need to do if you want to eliminate the mixed term here: http://www.sparknotes.com/math/precalc/conicsections/section5.rhtml . If you need to describe the ellipse further (location of center, length of axes, etc.), you will need to make the transformation first, then solve the standard form equation you obtain.

 

[chaoseverlasting]

If you want to convert the equation you have to the standard form of the equation of an ellipse, use the transformation

$$X=xcos\theta -ysin\theta$$

$$Y=xcos\theta +ysin\theta$$

And then equate the xy term to zero. This will give you theta which you substitute in the equation you get to get the standard form of the equation.

 

[Defennder]

Or if you know a little bit of linear algebra and orthogonal diagonolization you can easily express the equation as an ellipse. I got this as an answer (x and y are wrt to rotated coordinates):

x^2/(106/15) + (y-1/3)^2/(53/45) = 1

 

[HallsofIvy]

The intersection of two surfaces, in R³, is, in general, a curve in R³, which cannot be written as a single equation (that would be the equation of a surface). What you need are parametric equations of the curve of intersection.

You talk about the projection into the xy-plane being an ellipse but I see nothing in the statement of the problem that asks for such a projection.

One method of solving this would be to solve the linear equation for one of the variables, say z= 2x+ y- 2. Now put that back into the first equation so you get a (quadratic) equation in x and y. Solve that for one variable, say x. You can now use x itself as the parameter, writing both y and z in terms of x.