Let $$A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$ be a matrix with determinant $1$. Then one can see that the conic given by the equation $$Q(x,y)=cx^2+(d-a)xy-by^2=C,\quad C\geq 0$$ is invariant under multiplication by $A$.
This comes from the fact that if $$M=\begin{pmatrix}c&d\\ -a&-b\end{pmatrix}$$ then $Q(v)=v^TMv$ and $A^TMA=M$.
I didn't succeed in finding resources on the subject. Does anyone know more on this subject and can point me to something of interest?