JEE Advanced 2021 · Paper 2, Mathematics question 14

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Let
$$M = \{(x, y) \in \mathbb R \times \mathbb R : x^2 + y^2 \le r^2\},$$
where $r > 0$. Consider the geometric progression $a_n = \dfrac1{2^{n-1}}$, $n = 1, 2, 3, \ldots$. Let $S_0 = 0$ and, for $n \ge 1$, let $S_n$ denote the sum of the first $n$ terms of this progression. For $n \ge 1$, let $C_n$ denote the circle with center $(S_{n-1}, 0)$ and radius $a_n$, and $D_n$ denote the circle with center $(S_{n-1}, S_{n-1})$ and radius $a_n$.