Equation for Bivariate Normal Density Ellipse

[#1]

Is there a way to determine or retrieve the equation of the density ellipse which is plotted in the Bivariate platform?

I tried to generate these ellipses manually using the eigenvalues and eigenvectors of the covariance matrix, but it does not seem to match those plotted by JMP.  Is this spelled out in the documentation somewhere?  Is there some other source which describes how JMP generates these ellipses?

[#2]

so a little history on how its calculated (really good old JMP stuff that seems to be missing from the current documentation)

http://biostat.jhsph.edu/courses/bio653/misc/JMPer%20Cable%20Summer%2098%20Why%20is%20it%20called%20...

Then there is Craige's blog on how to do it manually:  ( Kudos to @Craige_Hales )

https://community.jmp.com/t5/Uncharted/Ellipse/ba-p/21242

hopefully this helps, a little...