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iamaditya

Member

From: Planet Mars

Registered: 2016-11-15

Posts: 821

Difference between parabola and hyperbola

Can anyone tell me the difference between a parabola and hyperbola?

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Difference between parabola and hyperbola

hi iamaditya

Do you know the ellipse?  The parabola sits on the border between the ellipses and the hyperbolas. 

An ellipse is a single closed curve and the hyperbola is always open and has two disjoint parts.

This page: http://www.mathsisfun.com/geometry/conic-sections.html  tells about the conic sections.

Start with a double cone (that is a cone and another upside-down cone sitting on its vertex.

If you cut (make a section) with a horizontal plane the shape of the section is a circle.

Incline the plane a little and you get an ellipse.  As there are many angles for the cut there are many different ellipses you can make in this way.

Tilt the plane beyond the angle of slope of the cone and you get the 'family' of hyperbolas.  The planes cut the lower cone and the upper cone so you get two sections.  Again there are many angles that will result in hyperbolas.

When the angle of the cut equals the angle of the slope of the cone you get a parabola.  The cut can slice the lower (say) cone but will just never cut again on the top cone.  This is why I say it sits at the border between the hyperbolas and the ellipses.

Copernicus discovered that the planets orbit the Sun in elliptical orbits.  Newton showed mathematically why this is so.  It is possible for a planet to orbit a star in a circular orbit but in practice the orbit will always be at least slightly elliptical.

Comets that return also have elliptical orbits.  But it is possible for an object to enter the solar system and pass close to the Sun before swinging back out and never returning.   Such an orbit is a hyperbola.  Again is it possible for the orbit to be a parabola but it's not likely.

Search lights use reflecting mirrors that are parabolic.  If the bulb is at the focus of the parabola, then the light beam will produced parallel rays.  For the same reason astronomical reflecting telescopes have a parabolic surface so that all the light gathered is concentrated at the focus.

Bob

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