Lagrange Multipliers (Question)

Hello, I am trying to find the absolute maximum and/or minimum of

$\displaystyle f(x,y)=xy$ on the circle $\displaystyle x^2 + y^2=1$

I am stuck and would like some help if any1 can help me.

This is what I have so far,

$\displaystyle \nabla f(x,y)=\lambda (g(x,y))$

$\displaystyle \nabla f(x,y)=yi+xj$

$\displaystyle \nabla g(x,y)=\lambda (2xi+2yj)$

setting the components equal to each other:

$\displaystyle y=2x\lambda$

$\displaystyle x=2y\lambda$

constraint: $\displaystyle x^2 + y^2=1$

I am having trouble solving for $\displaystyle x, y, \lambda$

I have 3 equations (including constraint) and 3 unknowns, It should work out. Can anyone help me? This is an algebra question, I feel like a moron.