The plane $\displaystyle x+y+2z=2$ intersects the paraboloid $\displaystyle z=x^2+y^2$ in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

What I did:

Let $\displaystyle f(x,y)=x+y+2z-2$ and $\displaystyle g(x,y)=z-x^2-y^2$

$\displaystyle f_x=1,f_y=1,f_z=2$ and $\displaystyle g_x=-2x, g_y=-2y,g_z=1$

then:

$\displaystyle 1=-2x\lambda$

$\displaystyle 1=-2y\lambda$

$\displaystyle 2=\lambda$

$\displaystyle z=x^2+y^2$

$\displaystyle x=-1/4,y=-1/4,z=1/8$

But this is none of the answer. The answers are $\displaystyle (1/2,1/2,1/2) and (-1,-1,2)$

Can someone point out my mistake?