Eliminating $\theta$ between the two expressions

Question

How do we find the equation of this parametric curve $$2x=\cos {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos {\theta}\right)$$ $$2y=\sin {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos {\theta}\right)$$

I want to eleminate $\theta$ between the two equations and find $y$ in terms of $x$, but I don't see an obvious way to do this..

Wolfram alpha says this curve is a circle..

Answer 1

On division, $\tan\theta=\dfrac yx$

We have $$2x=\sqrt{\frac35}\cos\theta\sin\theta+\cos^2\theta$$

Dividing by $\cos^2\theta=\dfrac1{1+\tan^2\theta},$ $$2x(1+\tan^2\theta)=\sqrt{\frac35}\tan\theta+1$$

Can you find the destination from here?

Answer 2

Hint: $$ \begin{align} 2\sqrt{\frac35}y-2x+1 &=\frac35\sin^2(\theta)-\cos^2(\theta)+1\\ &=\frac85\sin^2(\theta)\\ &=\frac85\frac{\tan^2(\theta)}{1+\tan^2(\theta)}\\ &=\frac85\frac{y^2}{x^2+y^2} \end{align} $$