It is a simple question, but I haven't still had a course on this topic and I'm finding it hard to understand some basics.
Consider a $2\times2$ symmetric matrix over a field (for example $\mathbb{C}$); $A=\pmatrix{a&b\\ b&c}$ and define $D,\,T \in \mathbb{C}[a,b,c]$ as the determinant and trace of $A$. How do I prove that the ring of invariants under the orthogonal matrices is generated by $D$ and $T$?
EDIT
I'm afraid I messed up the question. Here is where the problems come from.
We represent the 2-degree algebraic curves on the plane as $3\times3$ symmetric matrices $\pmatrix{a&b&d\\ b&c&e\\ d&e&f}$ and we want to prove that the ring of invariants under the Euclidean transformation group $G$ is generated by $D,\,T$ and $E$, where $D$ is the determinant of the first $2\times2$ pricipal minor, $E$ is the determinant and $T$ the trace of the first $2\times2$ principal minor. In order to do this the text I am reading (S. MUKAI) considers the invariants under the normal subgroup of translations and proves that it is $\mathbb{C}[a,b,c,E]$. It then states that, being the quotient of $G$ over the translations $\mathrm{O}(2)$, the ring of invariants of $\mathbb{C}[a,b,c]$ is generated by $D$ and $E$.