Consider the circle $x^{2} + y^{2} = 25$ and a point A(1,2) lying inside it. Consider secants of the circle passing through the point A. It turns out that the midpoint of the secants, lie on another circle of centre (a,b) and radius r, then the value of $a + b + 2 r^{2} =$ ________

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answer is 4.

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Detailed Solution

Let $P (x_{1}, y_{1})$ be the midpoint of the chord
Equation of the chord is $S_{1} = S_{11}$
$\begin{array}{l} \Rightarrow {xx}_{1} + {yy}_{1} - 25 = x_{1}^{2} + y_{1}^{2} - 25 \\ \Rightarrow x_{1}^{2} + y_{1}^{2} - x_{1} x - y_{1} y = 0 \\ A (1, 2) \\ \Rightarrow x_{1}^{2} + y_{1}^{2} - x_{1} - 2 y_{1} = 0 \\ x^{2} + y^{2} - x - 2 y = 0 \\ C (\frac{1}{2}, 1), r = \frac{\sqrt{5}}{2} \\ a + b + 2 r^{2} = \frac{1}{2} + 1 + \frac{5}{2} = 4 . \end{array}$