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I've merely seen the hyperbola defined as the "set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant.".

Like here: https://people.richland.edu/james/lecture/m116/conics/hypdef.html

However, since the hyperbola is quite reminiscent of a "double" parabola, then is there also an algebraic link between the two?

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    $\begingroup$ Conic section. $\endgroup$ – Kaster Sep 16 '15 at 18:25
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    $\begingroup$ My favorite way of understanding the connection is through projective space: start with a circle, grab any point and start pulling/stretching, you will get an ellipse; keep going until the point reaches infinity, you get a parabola; slingshot around from infinity (now with a twist, since projective space is not orientable) and you get a hyperbola. You can also just think of this as tilting the double cone, which is of course the same thing. $\endgroup$ – AndrewG Sep 16 '15 at 18:41
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Both are conic sections: they can be obtained as the intersection of a cone and a plane. This results in the general expression of points on them as satisfying the equation $$ ax^2 + bxy + cy^2 + dx+ey+f=0; $$ if we have $b^2-4ac>0$, this describes a hyperbola, and if $b^2-4ac=0$, it is a parabola. (If $b^2-4ac<0$, it's an ellipse.) As a particular example, $$ cy^2+dx=0 $$ is a parabola if $c,d \neq 0$, and $$ax^2+cy^2+f=0$$ is a hyperbola if $ac<0$.

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Referring to the focus-directrix property of the conic, you could regard the parabola as the limiting case of the hyperbola where the eccentricity takes the value $1$ as opposed to a hyperbola where the eccentricity is $>1$

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