Question

Use rotation of axes to identify the curve: sqrt(3)*xy+y^2=1

Answer 1

Write as:
\[\vec{r}^TM\vec{r}=\sqrt3xy+y^2=1 \implies (x,y)\left[\begin{matrix}0 & \frac{\sqrt3}{2}\\ \frac{\sqrt{3}}{2} & 1\end{matrix}\right]\left(\begin{matrix}x \\ y\end{matrix}\right)\]
Now diagnolize M.