Proving Kepler's Law: A Math Challenge

[paintednails]

kepler's law!

hi, does anybody know how to prove that the eccentricity multiplied by the directrix is equal to [itex] \frac {b^2}{a} [\itex]?

i found that the eccentricity of an ellipse is equal to c/a.

i also found that the directrix is equal to a/e. the way i see it, if i multiply e and d, then i get

ed = a

but how do i prove that a = b^2 / a ?

please help, thanks <3

 

[HallsofIvy]

hi, does anybody know how to prove that the eccentricity multiplied by the directrix is equal to $\frac {b^2}{a}$?

i found that the eccentricity of an ellipse is equal to c/a.

i also found that the directrix is equal to a/e. the way i see it, if i multiply e and d, then i get

ed = a

but how do i prove that a = b^2 / a ?

please help, thanks <3

?? What do you mean by the directrix of an ellipse? A parabola has a directrix but it is a line, not a number.

 

[Office_Shredder]

I suspect we have an ellipse with center at the origin, major axis on the x-axis, minor on the y, foci at +/-c, major axis length a, minor length b, and vertical directrices at x=+/-a/e

Ifa=b²/a, then a²=b², which means you really have a circle.