Equation of auxiliary circle of ellipse $2x^2+6xy+5y^2=1$

Question

Equation of auxiliary circle of ellipse $2x^2+6xy+5y^2=1$

My attempt: I know that auxiliary circle means that it has the diameter of major axis

To remove I tried axis of rotation method and got $$\cot (2\theta)=-1/2 \\ x=x'\cos(\theta) -y'\sin(\theta) \\ y=x'\sin(\theta) + y'\cos(\theta)$$ And put this is equation I am getting somewhere wrong. Please help.

Answer 1

Using Principal values,

$$-\dfrac\pi2<2\theta<0,\cos2\theta=+\dfrac1{\sqrt{1+\tan^22\theta}}=?$$

$$\sin\theta=-\sqrt{\dfrac{1-\cos2\theta}2}=?$$

$$\cos\theta=\sqrt{\dfrac{1+\cos2\theta}2}=?$$