Different parametric functions give same result

Question

While trying to parametrise the intersection between $x^2+y^2=z^2$ and $z=\frac x2+2$, polar coordinates gave me $$r^2=4+2r\cos t+\frac14(r\cos t)^2$$ which returned two values of $r$: $$r=-\frac4{\cos t\pm2}$$ Using the form $\{r\cos t,r\sin t,\frac{r\cos t}2+2\}$, I was able to plot the intersecting shape. However I am still confused why both values of $r$ work and produce the same shape.