Let $ \varphi : GL_2 ( \mathbb{R} ) \times \mathcal{M}_2 ( \mathbb{R} ) \to \mathcal{M}_2 ( \mathbb{R} ) $ be the group action defined by : $ \varphi (P,A) = PAP^{T} $.
What does it mean that the conics of the form :
$$ ( \mathcal{C} ) \ \ : \ \ f(x,y) = ax^2 + bxy+cy^2 + dx + ey + f = 0 $$
form an invariant class by the group action $ \varphi $, and are divided into three orbits: ellipses, hyperboles, parabolas? How to show that ?
Thank you in advance for your help.