TITLE: Rotating a conic equation
QUESTION [0 upvotes]: Identify the type of conic, transform the equation in x and y into equation in X and Y (without an XY term) by rotating the x- and y- axes by $θ=30^\circ$ to arrive at the new X- and Y- axes and graph the resulting equation showing both sets of axes.
$7x^2 + 4\sqrt{3}xy + 3y^2 - 9 = 0$
EDIT: There was a typo. It is fixed

REPLY [0 votes]: This can be done very nicely in the complex plane. We first convert to polar form with $x=r\cos\theta,\,y=r\sin\theta$ so that
$$7x^2 + 4\sqrt{3}xy + 3y^2 - 9 = 0\to r^2(7\cos^2\theta+4\sqrt{3}\cos\theta\sin\theta+3\sin^2\theta)=9\\
r=\sqrt{\frac{9}{7\cos^2\theta+4\sqrt{3}\cos\theta\sin\theta+3\sin^2\theta}}$$
Then we can say
$$z=re^{i\theta},\quad \theta\in[0,2\pi]$$
And finally, any rotation, say $z_{\theta_0}$, is given by $z_{\theta_0}=ze^{i\theta_0}$.
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