Grrr... This problem has been stressing me out for the past half hour

Find the extreme values of f(x,y) = xy on the ellipse ((x^2)/8) + ((y^2)/2) = 1

i've gone ahead and figured out the gradient f and g and ended up with (we'll call L lambda since I don't know how to insert that

y = (Lx)/4

x = Ly

((x^2)/8) + ((y^2)/2) = 1

I've gotten it down to this system of equations but i'm having trouble solving for this. Any help is appreciate.