The equation of the curve is- r^2cos2(theta)=q^2
Sorry I don't know to write this in Natural form but I just hope that I am clear enough
That's even more confusing!
And what is "q"? Some constant?
Do you know the identity "$\displaystyle cos(2\theta)= cos^2(\theta)- sin^2(\theta)$"? You can use that to write your equation as $\displaystyle r^2cos^2(\theta)- r^2sin^2(\theta)= q$ and use the fact that $\displaystyle x= r cos(\theta)$, $\displaystyle y= r sin(\theta)$ to change to Cartesian coordinates. Looks like a hyperbola to me.