If a circle passes through the point (a, b) a | vedclass

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(B) Let the equation of the variable circle be ${x^2} + {y^2} + 2gx + 2fy + c = 0$ $... (i)$.Since circle $(i)$ cuts the circle ${x^2} + {y^2} - 4 = 0

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$2ax + 2by - ({a^2} + {b^2} + 4) = 0$

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(B) Let the equation of the variable circle be ${x^2} + {y^2} + 2gx + 2fy + c = 0$ $... (i)$.Since circle $(i)$ cuts the circle ${x^2} + {y^2} - 4 = 0$ orthogonally,the condition $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ gives $2g(0) + 2f(0) = c - 4$,which implies $c = 4$.Since the circle $(i)$ passes through the point $(a, b)$,we have ${a^2} + {b^2} + 2ga + 2fb + 4 = 0$.To find the locus of the centre $(-g, -f)$,let $x = -g$ and $y = -f$,so $g = -x$ and $f = -y$.Substituting these into the equation,we get ${a^2} + {b^2} + 2(-x)a + 2(-y)b + 4 = 0$.Thus,the locus is $2ax + 2by - ({a^2} + {b^2} + 4) = 0$.

If a circle passes through the point $(a, b)$ and cuts the circle ${x^2} + {y^2} = 4$ orthogonally,then the locus of its centre is

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