If a variable line 3 x + 4 y - λ = 0 is such that the two circles x 2 + y 2 - 2 x - 2 y + 1 = 0 and x 2 + y 2 - 18 x - 2 y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval :

Step-by-step solution

Given circles x 2 + y 2 - 2 x - 2 y + 1 = 0 ⇒ x - 1 2 + y - 1 2 = 1 2 and x 2 + y 2 - 18 x - 2 y + 78 = 0 ⇒ x - 9 2 + y - 1 2 = 2 2 Thus, centers of two circles are 1 ,   1   &   9 ,   1 and radii are 1   &   2 respectively. Since, center of circles are on the opposite side of line 3 x + 4 y - λ = 0 . ⇒ 3 + 4 - λ 27 + 4 - λ < 0 ⇒ 7 - λ 31 - λ < 0 ⇒ λ ∈ 7 ,   31   … … . i Since, circl