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(B) The equation of any circle passing through the intersection of circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 + \lambda S_2 = 0$.Thus,the equati

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$x^{2} + y^{2} + 22y + 9 = 0$

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(B) The equation of any circle passing through the intersection of circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 + \lambda S_2 = 0$.Thus,the equation of the required circle is:$(x^{2} + y^{2} - 8x - 2y + 7) + \lambda (x^{2} + y^{2} - 4x + 10y + 8) = 0 \dots (i)$Rearranging the terms:$x^{2}(1 + \lambda) + y^{2}(1 + \lambda) - x(8 + 4\lambda) - y(2 - 10\lambda) + (7 + 8\lambda) = 0$Dividing by $(1 + \lambda)$:$x^{2} + y^{2} - x\left(\frac{8 + 4\lambda}{1 + \lambda}\right) - y\left(\frac{2 - 10\lambda}{1 + \lambda}\right) + \frac{7 + 8\lambda}{1 + \lambda} = 0$The center of this circle is $\left(\frac{4 + 2\lambda}{1 + \lambda}, \frac{1 - 5\lambda}{1 + \lambda}\right)$.Since the center lies on the $y-$axis,the $x-$coordinate of the center must be $0$:$\frac{4 + 2\lambda}{1 + \lambda} = 0 \implies 4 + 2\lambda = 0 \implies \lambda = -2$.Substituting $\lambda = -2$ into equation $(i)$:$(x^{2} + y^{2} - 8x - 2y + 7) - 2(x^{2} + y^{2} - 4x + 10y + 8) = 0$$x^{2} + y^{2} - 8x - 2y + 7 - 2x^{2} - 2y^{2} + 8x - 20y - 16 = 0$$-x^{2} - y^{2} - 22y - 9 = 0$$x^{2} + y^{2} + 22y + 9 = 0$.

Find the equation of the circle passing through the intersection of the circles $x^{2} + y^{2} - 8x - 2y + 7 = 0$ and $x^{2} + y^{2} - 4x + 10y + 8 = 0$,and having its center on the $y-$axis.

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