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Can someone tell me how is pair of straight lines a conic section. I know the equation is of second degree and other mathematical facts prove that. But how to visualise it? How is a pair of straight lines formed when a plane intersects a cone?

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    $\begingroup$ Take the plane to contain the axis of the cone. $\endgroup$ – user247327 Jul 21 '17 at 12:53
  • $\begingroup$ See Wikipedia's "Degenerate conic" entry, which describes not only the crossed lines case (where the plane contains the cone's axis) but the parallel lines case (where the cone itself degenerates into a cylinder). $\endgroup$ – Blue Jul 22 '17 at 14:29
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Sometimes an image is worth a thousand words: enter image description here

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"If a cone is cut by a plane through the vertex, the section is a triangle." (Apollonius, Conics, I, 3). Ignore the base of the triangle, which is a straight line within the circular base of the cone, and you have the two intersecting straight lines on the conic surface.

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You can think of this as a degenerate hyperbola, e.g., $$y=\lim_{a\to0}\pm\sqrt{a+x^2}$$

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Take the cone $$ z^2 = x^2 + y^2 $$ Intersect it with the $y = 0$ plane to get $$ z^2 = x^2 + 0^2 = x^2 $$ so that $$ z = \pm x $$ That's a pair of lines in the $y=0$ plane.

If you think of $z$ as "up and down", then $z^2 = x^2 + y^2$ is a double cone, and looks like an egg-cup or hourglass. You slice this with a vertical plane like $y = 0$, and you get a pair of the generating lines for the double-cone.

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