If you put $\displaystyle z = ce^{i\theta}$ (so that $\displaystyle z=x+iy$ lies on the circle $\displaystyle |z|=c$), then
$\displaystyle w = \frac12(z+z^{-1}) = \frac12(ce^{i\theta} + c^{-1}e^{-i\theta}) = \frac12(c+c^{-1})\cos\theta + \frac i2(c-c^{-1})\sin\theta.$
So w lies on the ellipse $\displaystyle \dfrac{x^2}{\bigl(\frac12(c+c^{-1})\bigr)^2} + \dfrac{y^2}{\bigl(\frac12(c-c^{-1})\bigr)^2} = 1.$