We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. We equip the vector space $\mathbb { R } ^ { 3 }$ with its canonical Euclidean affine structure and its canonical frame. For every matrix $A$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$, we denote by $\mathcal { Q } _ { A }$ the set of points of $\mathbb { R } ^ { 3 }$ whose image by the application $j$ has the same characteristic polynomial as $A$. a) Let $r$ be a strictly positive real number. Show that each of the sets $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { r J _ { 0 } }$ and $\mathcal { Q } _ { r H _ { 0 } }$ is a quadric for which an equation will be specified. b) Draw graphically the appearance of the quadrics $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { J _ { 0 } }$ and $\mathcal { Q } _ { H _ { 0 } }$ on the same drawing.
We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. We equip the vector space $\mathbb { R } ^ { 3 }$ with its canonical Euclidean affine structure and its canonical frame. For every matrix $A$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$, we denote by $\mathcal { Q } _ { A }$ the set of points of $\mathbb { R } ^ { 3 }$ whose image by the application $j$ has the same characteristic polynomial as $A$.
a) Let $r$ be a strictly positive real number. Show that each of the sets $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { r J _ { 0 } }$ and $\mathcal { Q } _ { r H _ { 0 } }$ is a quadric for which an equation will be specified.
b) Draw graphically the appearance of the quadrics $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { J _ { 0 } }$ and $\mathcal { Q } _ { H _ { 0 } }$ on the same drawing.