Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ the eigenvalues of $A$. We consider the set $\Gamma \subset \mathbb { R } ^ { 2 }$ defined by the equation $\langle A X , X \rangle = 1$. II.B.1) Characterize the conditions on the $\lambda _ { i }$ for which this set is: a) empty; b) the union of two lines; c) an ellipse; d) a hyperbola. II.B.2) Represent on the same figure the sets $\Gamma$ obtained for $A$ diagonal with $\lambda _ { 1 } \in \{ - 4 , - 1,0,1 / 4,1 \}$ and $\lambda _ { 2 } = 1$.
Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ the eigenvalues of $A$.
We consider the set $\Gamma \subset \mathbb { R } ^ { 2 }$ defined by the equation $\langle A X , X \rangle = 1$.
II.B.1) Characterize the conditions on the $\lambda _ { i }$ for which this set is:
a) empty;
b) the union of two lines;
c) an ellipse;
d) a hyperbola.
II.B.2) Represent on the same figure the sets $\Gamma$ obtained for $A$ diagonal with $\lambda _ { 1 } \in \{ - 4 , - 1,0,1 / 4,1 \}$ and $\lambda _ { 2 } = 1$.