In indical notation, we can write equation of conic as:
$$ s_{ij} = Ax_i x_j + B(x_i y_j + x_j y_i) + C y_i y_j + F(x_i + x_j) + G(y_i + y_j) + H=0$$
Where $i$ and $j$ stands in for $(x_i, y_i)$ and $(x_j,y_j)$. Now my question is there, an algorithim for finding a $k$ such that,
$$ s_{ij} = s_{kk}$$
Like plugging in a pair of points is equal to a plugging a single point into the equation?
With blue's comment, I now realized, the correct question to ask is if there is a "natural point" which we may take.