<s>[INST] How to find major & minor axes of the ellipse $10x^2+14xy+10y^2-7=0$? 
What are major & minor axes of the ellipse: $10x^2+14xy+10y^2-7=0$ ? 

My trial:
from given equation:  $10x^2+14xy+10y^2-7=0$
$$10x^2+14xy+10y^2=7$$
$$\frac{x^2}{7/10}+\frac{xy}{7/14}+\frac{y^2}{7/10}=1$$
I know the standard form of ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
But the term of $xy$ is a bottleneck. 
From here I can't proceed. 
Can somebody please help me solve this problem? 
Thank you. 
 [/INST] Rewriting the equation in polar coordinates, with $x=r\cos\theta$ and $y=r\sin\theta$, we get
$$r^2(10\cos^2\theta+14\cos\theta\sin\theta+10\sin^2\theta)-7=0$$
Using the trig identities $\sin^2\theta+\cos^2\theta=1$ and $2\sin\theta\cos\theta=\sin2\theta$, we find this simplifies to
$$r^2(10+7\sin2\theta)-7=0$$
or
$$r=\sqrt{7\over10+7\sin2\theta}$$
The largest value of $r$ occurs when $\sin2\theta=-1$ and the smallest when $\sin2\theta=1$.  Thus the major axis is $2\sqrt{7/3}$ and the minor axis is $2\sqrt{7/17}$.
</s>