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## Homework Statement

"Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##.

Show that the intersection of a plane with this cylinder can be described by an equation of the form ##(αx+β)^2+y^2=C^2##"

## Homework Equations

##x^2+y^2=C^2##.

##z=Mx+B##

##(αx+β)^2+y^2=C^2##

## The Attempt at a Solution

If we assume the plane in question is perpendicular to the x,z plane (Which the book does seem to assume, going by the explanations preceding this question), it is of the form ##ax+bz=c##. Thus a point ##(x,y,z)## is on it iff ##z=Mx+B## for some ##M, B## (This is how Spivak described the plane so I'm going with this).

So a point ##(x,y,z)## belongs to the intersection iff

1) ##x^2+y^2=C^2##.

2) ##z=Mx+B##

Obviously right from the start I'm misunderstanding the question since all the equation ##(αx+β)^2+y^2=C^2## does is takes the original cylinder, squeezes (or stretches) it along the x axis and transports it along the x axis. Besides, since we can insert any z coordinate we want into a point described by this equation, it can't describe a 2D intersection.

I'm guessing that this equation is for after we "eliminate" the z axis and describe the intersection using a 2d coordinate system (of the intersecting plane), but I don't really know how to do that. I know that the new coordinate system would be:

##x = αx'+β, y=y'##

By this, it looks like all we need to do is insert these new coordinates in to the cylinder equation and call it a day, but like I said, something more is needed. What am I missing here?