The locus of the point 

$\left[\frac{\mathrm{cos\theta }\left(1-\cos \mathrm{\theta }\right)}{{\sin }^2\mathrm{\theta }}, \frac{1-\cos \mathrm{\theta }}{\sin \mathrm{\theta }}\right]$ 

represents a

$$\begin{gathered} \mathrm{Let} \mathrm{x},\mathrm{y}) \mathrm{be} \mathrm{locus} \mathrm{of} \mathrm{point} \\ \Rightarrow \mathrm{x}=\frac{\mathrm{cos\theta }(1-\mathrm{cos\theta })}{{\sin }^2\mathrm{\theta }} - (1) , \mathrm{y}=\frac{1-\mathrm{cos\theta }}{\mathrm{sin\theta }} - (2) \\ \Rightarrow \mathrm{x}=\frac{\mathrm{cos\theta }}{\sin \theta }\frac{(1-\mathrm{cos\theta })}{\mathrm{sin\theta }} \\ \Rightarrow \mathrm{x}=\mathrm{y} \mathrm{cot\theta } \\ \Rightarrow \cot \mathrm{\theta }=\frac{\mathrm{x}}{\mathrm{y}} \\ \mathrm{from} (2) \mathrm{y}=\frac{1-{\displaystyle \frac{\mathrm{x}}{\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}}}}{{\displaystyle \frac{\mathrm{y}}{\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}}}} \\ \Rightarrow {\mathrm{y}}^2=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}-\mathrm{x} \\ {\mathrm{y}}^4+{\mathrm{x}}^2+2{\mathrm{xy}}^2={\mathrm{x}}^2+{\mathrm{y}}^2 \\ \Rightarrow {\mathrm{y}}^2=1-2\mathrm{x} \\ \Rightarrow {\mathrm{y}}^2=-2\left(\mathrm{x}-\frac{1}{2}\right) \\ \Rightarrow a=\frac{1}{2} \\ Now \mathrm{focus}=(\mathrm{h}-\mathrm{a},\mathrm{k})=\left(\frac{1}{2}-\frac{1}{2},0\right)=(0,0) \end{gathered}$$