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enter image description here

If I were to show you this curve that is quite zoomed in is it possible to tell if it is a hyperbola of a parabola

This is a picture my friend sent me and I cannot tell which conic section it is.

Blue
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Linkin
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2 Answers2

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Draw two or more parallel chords (not perpendicular to the axis). If their midpoints are aligned parallel to the axis, then it's a parabola. Otherwise it's a hyperbola and the intersection between the line of the midpoints and the axis is its center.

enter image description here

Intelligenti pauca
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  • Is there a reason for this peculiar property or is it just something that "happens" – Linkin Nov 01 '20 at 10:56
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    @JustJohan It is a property of any conic section and follows from this theorem: if $O$ is the centre of a conic (the centre of a parabola being at infinity) and $T$ a point on the conic, then line $OT$ meets all chords parallel to the tangent at $T$ at their midpoints. This can be proved in many different ways. – Intelligenti pauca Nov 01 '20 at 12:33
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This looks like a hyperbola.

For $x > 4$, the gradients of the curve are nearly constant for both positive and negative values of $y$. This suggests it is approaching an asymptote, something which does not happen for a a parabola.

Toby Mak
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