91Ó°ÊÓ

Identify the quadric with the given equation and give its equation in standard form. $$x^{2}+y^{2}+z^{2}-4 y z=1$$

Identify the quadric with the given equation and give its equation in standard form. $$-x^{2}-y^{2}-z^{2}+4 x y+4 x z+4 y z=12$$

Identify the quadric with the given equation and give its equation in standard form. $$2 x y+z=0$$

Identify the quadric with the given equation and give its equation in standard form. $$16 x^{2}+100 y^{2}+9 z^{2}-24 x z-60 x-80 z=0$$

Identify the quadric with the given equation and give its equation in standard form. $$x^{2}+y^{2}-2 z^{2}+4 x y-2 x z+2 y z-x+y+z=0$$

Identify the quadric with the given equation and give its equation in standard form. $$\begin{array}{l} 10 x^{2}+25 y^{2}+10 z^{2}-40 x z+20 \sqrt{2} x+50 y+ \\ 20 \sqrt{2} z=15 \end{array}$$

Identify the quadric with the given equation and give its equation in standard form. $$\begin{array}{l} 11 x^{2}+11 y^{2}+14 z^{2}+2 x y+8 x z-8 y z-12 x+ \\ 12 y+12 z=6 \end{array}$$

Let \(A\) be a real \(2 \times 2\) matrix with complex eigenvalues \(\lambda=a \pm b i\) such that \(b \neq 0\) and \(|\lambda|=1 .\) Prove that every trajectory of the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) lies on an ellipse. [Hint: Theorem 4.43 shows that if \(\mathrm{v}\) is an eigenvector corresponding to \(\lambda=a-b i\), then the matrix \(P=[\operatorname{Re} v \quad \operatorname{Im} v]\) is invertible and \(A=P\left[\begin{array}{cc}a & -b \\\ b & a\end{array}\right] P^{-1} \cdot \operatorname{Set} B=\left(P P^{T}\right)^{-1} .\) Show that the quadratic \(\mathbf{x}^{T} B \mathbf{x}=k\) defines an ellipse for all \(k>0\) and prove that if \(x \text { lies on this ellipse, so does } A x .]\)