Let the mirror image of a circle c 1 : x 2 + y 2 - 2 x - 6 y + α = 0 in line y = x + 1 be c 2 : 5 x 2 + 5 y 2 + 10 g x + 10 f y + 38 = 0 . If r is the radius of circle c 2 , then α + 6 r 2 is equal to ______

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Let the mirror image of a circle c 1 : x 2 + y 2 - 2 x - 6 y + α = 0 in line y = x + 1 be c 2 : 5 x 2 + 5 y 2 + 10 g x + 10 f y + 38 = 0 . If r is the radius of circle c 2 , then α + 6 r 2 is equal to ______ Now Image of centre c 1 ≡ 1 , 3 in x - y + 1 = 0 is given by x 1 - 1 1 = y 1 - 3 - 1 = - 2 1 - 3 + 1 1 2 + 1 2 ⇒ x 1 = 2 , y 1 = 2 So, centre of circle c 2 ≡ 2 , 2 Now radius of circle c 2 will be 4 + 4 - 38 5 = 2 5 = r Now radius of   c 1 2 = radius of   c 2 2 as