Yes the lines within the Cartesian plane are infinite
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infinite number of lines
A cylinder has an infinite number of lines of symmetry (because a circle has an infinite number of lines of symmetry).
If it is a right circular cone, it has an infinite number of planes of symmetry. If it is an oblique circular cone, it has one plane of symmetry.
circle
I would say that there are an infinite number of planes that can pass through a pair of skew lines. In order to find the equation of a plane, all you need is three points. take two points off of one line and one point off of the other line and you should be able to derive the equation of a plane. Since the number of points on a line is infinite, an infinite number of planes can be derived.