How would you show that the polar equation #r=(ed)/(1+ecos(theta))# represents an ellipse if e<1, a parabola if e=1, or a hyperbola if e>1?

#r=(ed)/(1+ecos(theta))#

2 Answers
Mar 6, 2018

See below.

Explanation:

We have

#r+e r costheta = e d# but

#{(r cos theta = x),(r = sqrt(x^2+y^2)):}#

then

#sqrt(x^2+y^2) +e x = e d# or

#x^2+y^2 = e^2(x-d)^2# or

#f(x,y) = x^2+y^2 - e^2(x-d)^2=0# or

#f(x,y) = (x,y)((1-e^2,0),(0,1))((x),(y))+((2x)/d-1)d^2e^2=0#

The conic characterization is done according do the eigenvalues of

#((1-e^2,0),(0,1))# which are #{1-e^2,1}#

thus

#{(e < 1-> "ellipse - (both positive)" ),(e = 1 -> "parabola - (one null)"),(e > 1-> "hyperbola - (one positive other negative)"):}#

Mar 6, 2018

Please see below.

Explanation:

We know that a conic section is defined as locus of a point that moves so that ratio, known as #e#, of its distance from a fixed point called focus and a given line called directrix is always constant.

  1. in case this ratio is equal to one, it is a parabola,

  2. in case this ratio is less than one, it is an ellipse and

  3. in case this ratio is greater than one, it is a hyperbola.

Assume that in polar coordinates, focus is at pole and directrix is at a distance of #d# to the left of pole and perpendicular to polar axis.

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Now if #(r,theta)# is a point on the conic section, then its distance from focus is of course #r#. For distance between the point and directrix, we can split it in two parts, of which one is #d# and ther is #rcostheta#. Hence this distance is #d+rcostheta#

and equation of conic section is

#r/(d+rcostheta)=e#

or #r=ed+ercostheta#

or #r(1-ecostheta)=ed#

and #r=(ed)/(1-ecostheta)#

Observe that if directrix is to the right of focus, the equation of conic section would have been #r=(ed)/(1+ecostheta)#