Bivariate Contour Ellipse Area

I have found a research paper, where there is need to calculate the area of a bivariate contour ellipse. I am familiar with drawing a confidence ellipse, so my concern is about the formulas provided in this publication.
As it is mentioned the bivariate contour ellipse area is calculated with the following formula:

[math]area=2k\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^2}[/math] (1)
where [math]\rho[/math] is the product moment correlation coefficient and [math]\sigma_{x},\sigma_{y}[/math] are the standard deviations of the two variables.

It also states that k can be calculated through probability equation
[math]P=1-e^{^{-k}}[/math] (2)

I need help to understand which probability function is equation (2) and how we get as outcome for the area calculation equation (1)

Please help me if you are familiar with this
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