$If a circle C_{0} with radius 1 unit touches both the axes and as well as line (L_{1}) through P (0, 4) which cuts the x -axis at (x_{1}, 0) . Again a circle C_{1} is drawn touching x -axis, line L_{1} and another line L_{2} thorugh point P . L_{2} intersects x - axis at (x_{2}, 0) and this process is repeated n times. at (x_{2}, 0) and this process is repeated n times. The centre of circle C_{2} is (\frac{31}{k},1), hence k =$

Moderate

Solution

$\begin{array}{l} For C_{n} circle let centre be (a_{n}, 1) and it touches line L_{n} and L_{n + 1} \\ i.e. \frac{x}{x_{n}} + \frac{y}{4} = 1, \frac{x}{x_{n + 1}} + \frac{y}{4} = 1 \\ \Rightarrow \frac{4 α_{n} + x_{n} - 4 x_{n}}{\sqrt{16 + x_{n}^{2}}} = 1 \\ \Rightarrow \frac{4 α_{n} + x_{n + 1} - 4 x_{n + 1}}{\sqrt{16 + x_{n + 1}^{2}}} = 1 \end{array}$
$\Rightarrow 3 (x_{n + 1} - x_{n}) = \sqrt{16 + x_{n}^{2}} + \sqrt{16 + x_{n + 1}^{2}} \begin{array}{l} \Rightarrow \tan (\frac{θ_{n + 1}}{2}) = \frac{1}{2} \tan (\frac{θ_{n}}{2}) \\ Let x_{n} = 4 \cot θ_{n} 0 < θ_{n} < \frac{π}{2} As x_{1} = 3 \Rightarrow \tan \frac{θ_{1}}{2} = \frac{1}{2} \Rightarrow \tan (\frac{θ_{n}}{2}) = \frac{1}{2^{n}} \\ \Rightarrow x_{n} = (\frac{2^{2 n} - 1}{2^{n - 1}}) x_{2} = \frac{15}{2} \\ Puttingis (i), we get α_{2} = \frac{31}{4} \end{array}$

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