Vectors and conics
Vectors and conics

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Vectors and conics

4 Conics

4.1 Conic sections

Conic section is the collective name given to the shapes that we obtain by taking different plane slices through a double cone. The shapes that we obtain from these cross-sections are drawn below. It is thought that the Greek mathematician Menaechmus discovered the conic sections around 350 bc.

The circle in slice 7 can be regarded as a special case of an ellipse.

We use the term non-degenerate conic sections to describe those conic sections that are parabolas, ellipses or hyperbolas; and the term degenerate conic sections to describe the single point, single line and pair of lines.

Note: We usually use ‘conic’ rather than ‘conic section’ once we have described how conics arise.

There are some interesting features of the parabola, ellipse and hyperbola that we note for use later. The ellipse and the hyperbola both have a centre; that is, there is a point about which rotation through an angle is a symmetry of the conic. For example, for the ellipse and hyperbola illustrated above, the centre is the origin. On the other hand, the parabola does not have a centre. The hyperbola has two lines, called asymptotes, which it approaches.

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