<s>[INST] Find major and minor axis of given conic. Find the equation of major and minor axis of the given conic.Also find length of major and minor axis.
$$5x^2 + 5y^2 +6xy + 22x - 26y + 29$$ =0
I tried finding the nature of conic and centre of conic by partially differentiating it but could not think ahead.
 [/INST] Rewrite the equation using matrices:
$\begin{pmatrix}x&y\end{pmatrix}\begin{pmatrix}5&3\\3&5\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}22&-26\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+29=0$
$M=\begin{pmatrix}5&3\\3&5\end{pmatrix}, K=\begin{pmatrix}22&-26\end{pmatrix}$
Diagonalising $M$ using the eigenvalues and eigenvectors we get:
$P=\begin{pmatrix}\frac{\sqrt{2}}{2}&-\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{pmatrix}$
$\begin{pmatrix}x\\y\end{pmatrix}=P\begin{pmatrix}x'\\y'\end{pmatrix}, P^tMP=\begin{pmatrix}8&0\\0&2\end{pmatrix}$
$\begin{pmatrix}x'&y'\end{pmatrix}P^tMP\begin{pmatrix}x'\\y'\end{pmatrix}+KP\begin{pmatrix}x'\\y'\end{pmatrix}+29=0$
that is:
$8x'^2+2y'^2-2\sqrt{2}x'-24\sqrt{2}y'+29=0$
or completing the squares and rearranging:
$(\frac{x'-\frac{\sqrt{2}}{8}}{\frac18 \sqrt{922}})^2+(\frac{y'-6\sqrt{2}}{\frac14\sqrt{922}})^2=1$
Making the semi-major axis $\frac14\sqrt{922}$ and the semi-minor axis $\frac18 \sqrt{922}$.
Using the inverse transformation 
$\begin{pmatrix}x'\\y'\end{pmatrix}=P^t\begin{pmatrix}x\\y\end{pmatrix}:$
$x'-\frac{\sqrt{2}}{8}=0, \frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}y-\frac{\sqrt{2}}{8}=0$ 
we get the equation of one of the axes: $y=\frac14-x,$ and
$y'-6\sqrt{2}=0, -\frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}y-6\sqrt{2}=0$ 
the equation of the other axis $y=12+x.$
Edit:
In differentiating to get the center we get the following system:
$10x+6y+22=0$
$6x+10y-26=0$
Since by the comments we get a rotation by $\frac{\pi}{4}$ ($\tan(2\theta)=\frac{B}{A-C}$), let's try and find the two equations in this system that is rotated to that angle (adding and subtracting):
$16x+16y=4$
$-4x+4y=48$
These are the same two lines on the figure.

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