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Based from reading Math books I have this question, hope you can help me Sirs/Madams

Which is a better definition of a simplified parabola

  1. A locus of an equation $Cy^2+Dx=0$ or $Ax^2+Ey=0$. (In this definition, definition 2 then becomes a property of the given equations)

or

2.Set of all points in the plane equidistant from a fixed point called focus and a fixed line called directrix

Thanks a lot

Jr Antalan
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  • I like definition (2) personally, but I suppose others might like (1). – bjd2385 Jan 19 '15 at 22:44
  • It really depends on the context. If you are doing things in the coordinate plane (calculus), then (1) is definitely easier to work with. Otherwise if you are working in Euclidean geometry (straight-edge and compass), then (2) is easier to work with. – Mike Pierce Jan 19 '15 at 22:46
  • In definition 2 you should also state that the focus is not on the directrix. If it is, the locus is the line perpendicular to the directrix through the locus: not a parabola! – Rory Daulton Jan 19 '15 at 22:51
  • Thanks a lot, I learned a lot from your comments. – Jr Antalan Jan 20 '15 at 11:04

1 Answers1

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Definition 1 is not sufficient to characterize all parabolas in a coordinate plane. It characterizes some--namely, those whose axes of symmetry are coincident with one of the coordinate axes, but for parabolas whose axis of symmetry is not parallel to either coordinate axis, definition 1 is not sufficient.

Consequently, Definition 2 is preferable. There is no restriction on the directrix or focus so long as the latter is not a point on the former.

heropup
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