Question 939616
Given area of the ellipse to be 2pi
Let x and y be the adjacent sides of the ellipse as shown in the picture
pi*x*y = 12*pi
i.e. x*y = 12 -------eqn1

Also
Perimeter of ellipse is pi*(x+y)

Given that the distance between the vertices of x and y are 5units apart
This can form a right anglr triangle,
From pythagoras' theorem, {{{x^2+ y^2 = 5^2 = 25}}}

Also, to simplify things up here to get our perimeter,
We know that {{{(x+y)^2 = x^2+y^2+2*x*y}}}
SubStituting values
{{{(x+y)^2 = 25 + 2*12}}}
{{{(x+y)= sqrt(49)}}}
x+y = 7
But perimeter is pi * (x+y)
= 7 * pi

QED