Pair of straight lines as a conic section

Question

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?

Answer 1

Sometimes an image is worth a thousand words:

Answer 2

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.

Answer 3

"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.

Answer 4

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