A circle is concentric with the circle x^2 + | vedclass

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(A) The given circle is $x^2 + y^2 - 6x + 12y + 15 = 0$.Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we have $g = -3, f = 6, c = 15$.The radius $r_1

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$x^2 + y^2 - 6x + 12y - 15 = 0$

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(A) The given circle is $x^2 + y^2 - 6x + 12y + 15 = 0$.Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we have $g = -3, f = 6, c = 15$.The radius $r_1$ of the given circle is $\sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + 6^2 - 15} = \sqrt{9 + 36 - 15} = \sqrt{30}$.The area of the given circle is $A_1 = \pi r_1^2 = 30\pi$.The required circle is concentric,so its equation is $x^2 + y^2 - 6x + 12y + k = 0$.Its radius $r_2$ satisfies $r_2^2 = g^2 + f^2 - k = 45 - k$.The area of the required circle is $A_2 = \pi r_2^2 = \pi(45 - k)$.Given that $A_2 = 2A_1$,we have $\pi(45 - k) = 2(30\pi) = 60\pi$.$45 - k = 60 \Rightarrow k = -15$.Thus,the equation is $x^2 + y^2 - 6x + 12y - 15 = 0$.

$A$ circle is concentric with the circle $x^2 + y^2 - 6x + 12y + 15 = 0$ and has an area double that of the given circle. The equation of the circle is:

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