The question should be simple.

Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for $\lambda$.

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

$8x = \lambda2x$ ----> $\lambda = 4$

$20y = \lambda2y$ ---> $\lambda = 10$

$x^2 + y^2 = 4$

What am I missing?