Parabola vs Hyperbola, why does a Hyperbola have two foci/curves?

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• am4th
In summary, the eccentricity, e, of a conic section determines its shape and the number of foci and directrices it has. When e=1, we have a parabola with one focus and one directrix. As e increases, the shape becomes more elongated and we get a hyperbola with two foci and two directrices. The concept of a cone helps to visualize the different shapes, as the eccentricity represents the angle of the intersecting plane. However, the equations of these conic sections can also be understood without the use of a cone.

am4th

So I read a description saying something along the lines of, a Parabola does have a 2nd focus and directrix, but that they stretch off into infinity, whereas for the hyperbola the 2nd focus comes back round..?

Anyway, I'm trying to picture it and understand in relation to the eccentricity, e. What is it about e=1 (value for a parabola), that determines there is only one focus, directrix and curve as opposed to a hyperbola, e>1, that means there are two foci, directrices and curves.?

I get the feeling I'm missing something obvious..

Cheers!

Edit:

So I think I need to clarify my question a bit. My understanding is that a circle, ellipse, parabola and hyperbola are all specific cases of the same formula. The ratio between the the radius and the distance to the 'directrix' is called the eccentricity. When e =1, we have a parabola. When e>1 then we have a hyperbola. What I don't understand is, what happens when e>1 to lend the hyperbola the property of having two foci and two directrices, when at e=1 we had a parabola with only 1 focus and 1 directrix.

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It is always best to think of those as conic sections. In the pictures here and here, you can see how the center of a circle is torn into two which walk away from each other. First the ellipse, then the parabola and finally the hyperbola when it comes back from infinity and shows up on the other end of the double cone.

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fresh_42 said:
It is always best to think of those as conic sections. In the pictures here and here, you can see how the center of a circle is torn into two which walk away from each other. First the ellipse, then the parabola and finally the hyperbola when it comes back from infinity and shows up on the other end of the double cone.

Hey, thanks. Yes, it's much easier to understand from that perspective; however, I'm keen to understand it in relation to eccentricity. Sorry I didn't make this clear in the original post.

In my mind, whilst they can have a corresponding cone, a parabola/hyperbola can exist without a cone. The way I see it, right now I'm not studying the equation of a cone, I'm learning about equations of lines that can be described in the same way as the intersection of a plane and a cone. (I think that is correct..?)

I hope that makes sense.

am4th said:
Hey, thanks. Yes, it's much easier to understand from that perspective; however, I'm keen to understand it in relation to eccentricity. Sorry I didn't make this clear in the original post.

In my mind, whilst they can have a corresponding cone, a parabola/hyperbola can exist without a cone. The way I see it, right now I'm not studying the equation of a cone, I'm learning about equations of lines that can be described in the same way as the intersection of a plane and a cone. (I think that is correct..?)

I hope that makes sense.
You are correct. However you need to look at the cone as two cones meeting at at point. Plane, forming circle, ellipse, and parabola, intersecting cone crosses only one part. Hyperbola - plane crosses both parts.

HallsofIvy