graph quadratic form and find the equation of asymptotes

Question

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes.

$$ \left\{ \begin{aligned} 4(x_1)^2-12(x_1)(x_2)-(x_2)^2= 4 \end{aligned} \right. $$

I think it would be a rotated hyperbola, but I don't know exact how to graph it. And for the asymptotes, I can't transfer the equation into the form of $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Answer 1

Hint

Quoting Wikipedia :

To rotate a figure counterclockwise around the origin by some angle $\theta$ is equivalent to replacing every point with coordinates $(x,y)$ by the point with coordinates $(X,Y)$, where $$x=X \cos (\theta )-Y \sin (\theta )$$ $$y=X \sin (\theta )+Y \cos (\theta )$$

So, transform accordingly your equation $4x^2-12xy-y^2=4$; identify the $XY$ term and say that its coefficient is $0$. This should give you one equation for $\theta$. Solve it and replace $\sin(\theta)$ and $\cos(\theta)$ by their resulting values. The remaining should be simple.

Answer 2

The asymptotes correspond to the roots of the quadratic polynomial in $x_1,x_2$, as the variables will strive to "achieve $0=4$".

By the classical formulas,

$$x_1=\frac{6\pm2\sqrt{10}}{4}x_2.$$

The rotated axis are the bisectrices.