Reduce to canonical form and identify…

Reduce to canonical form and identify: $5x^2+8y^2+5z^2+4xy-8xz+4yz-27=0$.

Explanation of Method

A quadric with a zero eigenvalue and a double eigenvalue: a circular cylinder.

\[ \det(A-\lambda I)=-\lambda(\lambda-9)^2,\qquad \lambda_1=0\ \ \{2,-1,2\},\quad \lambda_{2,3}=9 . \]

The eigenspace for $9$ is the plane $2x-y+2z=0$. An orthonormal basis of it is $\tfrac1{\sqrt5}(1,2,0)$ and $\tfrac1{3\sqrt5}(-4,2,5)$.

\[ 9\eta^2+9\zeta^2=27 . \]

\[\eta^2+\zeta^2=3\]

This is a circular cylinder of radius $\sqrt3$. Its axis is the line through $O$ with direction $\{2,-1,2\}$.

Direct check: the equation is $9\bigl(x^2+y^2+z^2\bigr)-(2x-y+2z)^2=27$. That says

\[ |OM|^2-\left(\tfrac{2x-y+2z}3\right)^2=3 , \] that is, the squared distance from $M$ to the axis equals $3$. $\checkmark$

$O$ $\{2,-1,2\}$

The circular cylinder of radius $\sqrt3$ whose axis is the line through $O$ with direction $\{2,-1,2\}$.

\[\boxed{\,\eta^2+\zeta^2=3\,}\]