JEE Advanced 2021 Paper 2 Q14 Mathematics Coordinate Geometry Circles Hard

JEE Advanced 2021 Paper 2 · Q14 · Circles

Consider $M$ with $r = \dfrac{(2^{199}-1)\sqrt{2}}{2^{198}}$. The number of all those circles $D_n$ that are inside $M$ is

  1. A. $198$
  2. B. $199$
  3. C. $200$
  4. D. $201$
Reveal answer + step-by-step solution

Correct answer:B

Solution

$D_n$ has center $(S_{n-1}, S_{n-1})$ and radius $a_n = \dfrac{1}{2^{n-1}}$. $D_n\subset M$ requires $\sqrt{2}S_{n-1} + a_n \le r$, i.e. $\sqrt{2}\cdot 2(1 - 2^{-(n-1)}) + 2^{-(n-1)} \le \dfrac{(2^{199}-1)\sqrt{2}}{2^{198}}$. Simplifying: $2\sqrt{2} - 2^{-(n-1)}(2\sqrt{2}-1) \le 2\sqrt{2} - \dfrac{\sqrt{2}}{2^{198}}$, so $2^{-(n-1)}(2\sqrt{2}-1) \ge \dfrac{\sqrt{2}}{2^{198}}$. This gives $2^{2n-3} \le 2(2\sqrt{2}-1)^2/2 \approx 2^{397}$, leading to $n \le 199$. Hence 199 circles. Option (B).

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