Determine and classify the projective conic that contain the points:
$\langle 0,0,1\rangle,\langle 0,1,1\rangle,\langle 1,0,1\rangle,\langle 1,1,1\rangle,\langle 1 / 2,2,1 \rangle$.
I used this projective conic equation but then I can't reach any conclusion:
F(x,y,z) = ax^2 +by^2 + cz^2 + dxy + exz + fyz
By substituting the values of the points I obtained the following equation:
-8fx^2 - fy^2 + 8fxz + fyz = 0
I can't develop further. How to resolve this?
I will probably have to calculate the projective extension, that is, solve the conic at infinity, when z=0. In this case I get an ellipse because x=y=z=0, that is, there are no points at infinity. Is that it?