What are some other ways in which a parabola is "between an ellipse and a hyperbola"?
On page 122 of Gilbert Strangs calculus text he writes:
"Throughout mathematics, parabolas are on the border between ellipses and hyperbolas."
Here are three ways in which we can think of a parabola as being inbetween an ellipse and a hyperbola:
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If we cut a cone with a horizontal plane, we get a circle. When we tilt the plane slightly, we get an ellipse. If we tilt the plane a lot, we get a hyperbola. When we tilt the plane so that its angle matches the slope of the cone, we get a parabola.
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The equation $Ax^2+Bxy+Cy^2=1$ produces a hyperbola if $B^2 > 4ac$, an ellipse if $B^2-4AC <0$ and a parabola if $B^2-4ac=0$
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In the polar form $r=\frac\ell{1+e\cos\theta}$, we pass smoothly through a parabola when the eccentricity $e$ passes through $1$ with a fixed semi latus rectum $\ell$.
I'm suspecting there is at least one more way to understand why we think of a parabola as being inbetween an ellipse and a hyperbola, perhaps in terms of foci. Strang writes that the "second foci of a parabola" is located at ininfinity, and I'm not quite sure why this type of thinking makes sense and if we can somehow relate this foci at infinity to being an inbetween case of defining ellipses and hyperbola by their foci.
Best Answer
If you've ever used Kerbal Space Program or other spaceflight simulator (or work for NASA as an actual rocket scientist), you'll notice that your rocket's velocity around a planet will determine the shape of its projected orbit. Sometimes you'll get an elliptical orbit. Sometimes you'll get a hyperbolic "orbit". If you arrange things just right, you can get a parabolic orbit, and that speed (known as escape velocity) will be exactly at the transition point between elliptical-orbit and hyperbolic-orbit speeds.