Find the radical centre of the sets of circles ${(x - 2)}^{2} + {(y - 3)}^{2} = 36$ , ${(x + 3)}^{2} + {(y + 2)}^{2} = 49$ , and ${(x - 4)}^{2} + {(y + 5)}^{2} = 64$ .

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Solution

If we consider three circles as $S_{1}, S_{2}, S_{3}$ then for radical center, $S_{1} - S_{2} = 0 a n d S_{2} - S_{3} = 0.$
For radical center a point $(x, y)$ should be there such that,
$(x - 2)^{2} + (y - 3)^{2} - (x + 3)^{2} - (y + 2)^{2} = 36 - 49..... (1)$
and, $(x + 3)^{2} + (y + 2)^{2} - (x - 4)^{2} - (y + 5)^{2} = 49 - 64....... (2)$
Solving (1) and (2),we get
$x = \frac{26}{25}$
$y = \frac{13}{50}$

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(x - 1)^{2} + (y - 2)^{2} = 341

(x - 4) + (y - 1)^{2} = 341, (x - 5)^{2} + (y - 4)^{2} = 341

2 x - 3 y = 10, 2 x - 3 y = 4

2 x + 3 y = 1,

{(x - 3)}^{2} + {(y - 2)}^{2} = 9, {(x + 5)}^{2} + {(y + 6)}^{2} = 9

x^{2} + y^{2} = 25

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Find the centre and radius of the circles.
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