Asymptotes and focus of a conic? This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$
$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\right),
$$
His characteristic polynomial is: $p_A(\lambda) = \lambda^2-2\lambda-8$
A has eigenvalues discordant $(\lambda = 4, \lambda = -2)$, so it's an Hyperbole.
Then i found that the center of the conic is: $(-\frac{1}{16}, -\frac{5}{16})$
Then with the eigenvalues i found the two lines passing through the center:
$$4x-4y-1=0$$
$$8x+8y+3=0$$

Now i want to find the focus and the asymptotes but i have no idea how to do it.There is a way to find These two things through the data I have now? or do i need the canonical form of the conical? Thanks

You've two principal axes:
\begin{align*}
  x^2+6xy+y^2+2x+y+\frac{1}{2} & \equiv
 A\left( \frac{4x-4y-1}{\sqrt{4^2+4^2}} \right)^2+
 B\left( \frac{8x+8y+3}{\sqrt{8^2+8^2}} \right)^2+C \\
 & \equiv
 -2\left( \frac{4x-4y-1}{4\sqrt{2}} \right)^2+
 4\left( \frac{8x+8y+3}{8\sqrt{2}} \right)^2+\frac{9}{32}
\end{align*}
Now
\begin{align*}
  \frac{4x-4y-1}{4\sqrt{2}} &= x'\\
  \frac{8x+8y+3}{8\sqrt{2}} &= y'\\
  a &= \frac{3}{8} \\
  b &= \frac{3}{8\sqrt{2}} \\
  \frac{x'^2}{a^2}-\frac{y'^2}{b^2} &=1
\end{align*}
Asymptotes
$$b x' \pm a y'=0$$
Foci
$$(x',y')=(\pm \sqrt{a^2+b^2},0)$$
