Find the sum of co-ordinates of the centre of the smallest circle touching the circles $x^{2} + y^{2} - 2 y - 3 = 0$ and $x^{2} + y^{2} - 8 x - 18 y + 93 = 0$

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Solution

REF.Image.
$C_{1} = x^{2} + y^{2} - 2 y - 3 = 0$
$C_{1} = (0, 1) r_{1} = b \sqrt{1^{2}} + 3 = 2$
$C_{2} = (x^{2} + y^{2} - 8 x - 18 y + 93 = 0$
$C_{2} = (4, 9) = r_{2} = \sqrt{4} = 2$
$C_{1} C = r_{1} + r = 2 + r$
$C_{2} C = r_{2} + r = 2 + r$
C = mid pt of $C_{1} & C_{2}$
$= (\frac{0 + 4}{2}, \frac{1 + 9}{2}) = (2, 5)$

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