Consider a circle C 1   :   x   2 + y 2   –   4 x   –   2 y   =   α   –   5 . Let its mirror image in the line y = 2 x + 1 be another circle C 2 : 5 x 2 + 5 y 2   – 10 f x   –   10 g y +   36   =   0 . Let r be the radius of C 2 . Then α +   r is equal to _ _ _ _ _ _ _ _

Step-by-step solution

Given, C 1   :   x 2 +   y 2 – 4 x –   2 y + 5   –   α =   0 So, its centre will be, O 1 = 2 ,   1 and radius = α And C 2   :   5 x 2 + 5 y 2 – 10 f x – 10 g y + 36 = 0 ⇒ C 2   :   x 2 + y 2 – 2 f x – 2 g y   + 36 5 = 0 So, Centre O 2   =   f ,   g and radius r = f 2 + g 2 - 36 5 Also given O 2 is reflection of O 1 in 2 x – y + 1 = 0 , so image formula we get, ⇒ f - 2