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Consider the family \[C_{\alpha}:\big({2(\alpha-\cos{t})\,\cos{t},\,2(\alpha-\cos{t})\,\sin{t}}\big)\,,\;t\in[0,2\pi]\,,\;\alpha\in\mathbb{R},\] of closed parametric curves. Let $X_{\alpha}$ is the image set of $C_{\alpha}$ equipped with the induced topology of $\mathbb{R}^2$.

1. For which values of $\alpha$ these topological spaces $X_{\alpha}$ are homotopy-equivalent to topological space $X_{0.5}$?
2. For those spaces which are homotopy-equivalent to $X_{0.5}$ provide a homotopy equivalence.

In the figure below depicted $6$ curves of the family $C_{\alpha}$ for $\alpha\in\{{0.3,0.5,0.6,1.0,1.3}\}$.