Cylindrical section is an ellipse?

[uman]

Prove or disprove: The intersection of the plane $$x+y+z=1$$ and the cylinder $$x^2+y^2=1$$ is an ellipse.

 

[olliemath]

If you perform a change of variables so that the x+y+z=1 plane is the new 'x-y plane' then do you not get a new equation for your cylinder? You can then work it out by setting the new 'z' to zero?

 

[HallsofIvy]

The intersection of the two figures is the values of (x, y, z) that satisfy both equations simutaneously so solve the two equations simultaneously: From the first equation, y= 1- x- z. What do you get when you replace y by that in your second equation?

 

[uman]

Thanks for the help... I actually found a (more general) proof in a geometry textbook (Geometry and the Imagination by David Hilbert)

 

[chaoseverlasting]

It would have to be an ellipse because the plane isn't parallel to the cylinder. The normal vector is i+j+k which makes an angle of $$cos\alpha =\frac{1}{\sqrt3}$$ with all three axes.