I need to rotate $9x^2-2\sqrt{3}xy+7y^2=10$ such that it becomes into $3x^2+5y^2=5$

Question

I need to rotate $9x^2-2\sqrt{3}xy+7y^2=10$ such that it becomes into $3x^2+5y^2=5$.

Answer 1

Let $x=X \cos \theta + Y \sin \theta$ and $y= -X \sin \theta + Y \cos \theta $ in your quadratic. Then set the coefficient of $X Y$ to zero to get the angle of rotation $\theta$. Your quadratic would become $aX^2+bY^2=1.$ This proceedure is called orthogonalization.