Find the maximum and minimum value of the expression:
(y^2)/121 + (w^2)/81
if x, y, z, satisfy the following system:

x^2 + y^2 - 8x + 6y - 96 = 0
z^2 + w^2 + 10z - 4w - 52 = 0
xw + yz - 2x - 4w + 5y + 3z - 76 > 0

So far I have done the following:

x^2 -8x + y^2 + 6y=0
(x-4)^2 + (y+3)^2=121

z^2 + 10z + w^2 - 4w=52
(z+5)^2 + (w-2)^2=81

xw-2x+yz+3z-4w+5y>76
x(w-2) + z(y+3) - 4w + 5y>76

Where do I go from here?

I believe you have to use Lagrangians to solve such problems.

I think the following web site will help:

http://www.math.colostate.edu/~butle..._lecture14.pdf