The equation of the circle concentric with the circle x² + y² – 6x + 12y + 15 = 0 and double its area is __________.

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Solution

Given circle has equation
$x^{2} + y^{2} - 6 x + 12 y = - 15 i . e . x^{2} - 6 x + y^{2} + 12 y + 15 = 0 i . e . x^{2} - 2 (3) x + 9 - 9 + y^{2} + 2 y (6) + 36 - 36 + 15 = 0 i . e . {(x - 3)}^{2} + {(y - 6)}^{2} - 45 + 15 = 0 i . e . {(x - 3)}^{2} + {(y - 6)}^{2} = 30 ∴ Area of given circle is 30 π sq units$
Hence,
Given:- New circle has same center (3, −6) and area equal to double of 30π sq. unit.
Let r denote the radius of new circle
Hence πr² = 2(π 30)
i.e r² = 60
Hence, equation of new concentric circle is

${(x - 3)}^{2} + {(y + 6)}^{2} = 60 i . e . x^{2} - 6 x + 9 + y^{2} + 12 y + 36 = 60 i . e . x^{2} + y^{2} - 6 x + 12 y - 15 = 0$

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