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Find the centre of the circle passing through (0,0) and (1,0) and touching the circle x²+y² = 9.
Let the equation of circle isx² + y² + 2gx + 2fy + c = 0centre = (-g,-f)radius = √(g²+f²-c)it passes through (0,0)&(1,0) so point satisfy the equation put (0,0)c = 0put (1,0)g = -1/2since the circle touches internally hencedistance between centre = difference of radiuscentre of given circle = (0,0)radius = 3√(g² + f²) = 3 - √(g²+f²-c)√(1/4 + f²) = 3 - √(1/4 + f²)√(1/4 + f²) = 3/21/4 + f² = 9/4f² = 2f = √2 or -√2hence centre = (1/2,√2) or (1/2,-√2)
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