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March 29, 2024, 06:58:39 am

Author Topic: What are the conic sections?  (Read 1163 times)  Share 

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frog1944

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What are the conic sections?
« on: April 18, 2017, 12:49:57 pm »
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Hi,

I know that a circle, ellipse, parabola and hyperbola are conic sections, but is a point, and 2 lines intersecting also considered a conic section?
If not in the HSC syllabus, what about in "proper" conics?

Thanks

RuiAce

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Re: What are the conic sections?
« Reply #1 on: April 18, 2017, 01:28:40 pm »
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Short answer: No to both.

Long answer:
Firstly, conics in the HSC syllabus is still "proper" conics. It's just taught in a useless manner.

Loosely speaking, the ellipse, parabola and hyperbola categorise the main type of conic sections that we encounter. The types of conic sections you have mentioned are what's known as 'degenerate cases'. Note that a circle is just a special case of the ellipse, where the two foci coincide with the centre.

Recall that if we want to produce each conic section, we must slice the double cone accordingly
- Ellipse: At an angle less than the angle of inclination (of the side of the cone).
- Parabola: At an angle equal to the angle of inclination
- Hyperbola: At an angle exceeding the angle of inclination

The point is a special case of the ellipse. It occurs when we introduce the slice through the intersection between the two cones.

The single line is a special case of the parabola. It occurs when we slice exactly along the edge of the cone.

The pair of intersecting lines is a special case of the hyperbola. It also occurs when we slice through the intersection between the cones.

Note additionally, that the defining characteristic of a conic section is \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\). For these degenerate conic sections, we always have \(A=B=C=0\) (although the conditions on D, E and F are still variable).

So in a way, sort of. They could be considered separate conic sections, but they really fall into the umbrella of a broader category of conic sections.

frog1944

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Re: What are the conic sections?
« Reply #2 on: April 18, 2017, 08:13:57 pm »
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Wow! That makes sense. Thanks for the great answer RuiAce :)