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Let be $ f=(d,0) $, with $d >0 $ a point beeing in euclidean level.

For $ a > 0 $ is $ D_a \subset \mathbb{R}^2 $ the set of points $p =(x,y) \in \mathbb{R}^2 $

with $ || p-f|| = ax $

How can I show that $ D_a $ is for

$ a< 1 $ an ellipse

$a=1 $ a parabola

$a>1 $ a hyperbola

Any help very appreciated, I don't really know how to persue this proof.

wondering1123
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1 Answers1

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What you have stated is actually the definition of eclipse, parabola and hyperbola.

Basically,

$ \frac{\lVert p-f \rVert}{x} = a$ (ratio of distance from focus to directrix)

where f is focus and y-axis is directrix. (x is distance from y-axis)

Now the definition says, the ratio is:

= 1 for parabola,

< 1 for ellipse,

> 1 for hyperbola.

also, this ratio is called the eccentricity of the curve.

dumbguywithmathsmajor
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  • okay, I did not know about this definition..can you maybe state a link? I can't find information on this definition. – wondering1123 Jun 22 '22 at 08:10
  • "The locus of a point P, which moves so that its distance from a fixed point is always in a constant ratio to its perpendicular distance from a fixed straight line, is called a Conic Section." This is how S.L. Loney defines a conic section, now depending on the value of the ratio, it is categorised as parabola, ellipse, and hyperbola. You can have a look at the book, elements of coordinate geometry by SL Loney – dumbguywithmathsmajor Jun 22 '22 at 08:29