The set of values of k for which the circle C : 4 x 2 + 4 y 2 - 12 x + 8 y + k = 0 lies inside the fourth quadrant and the point 1 , - 1 3 lies on or inside the circle C is

Step-by-step solution

Given equation of circle C : 4 x 2 + 4 y 2 - 12 x + 8 y + k = 0 ⇒ x 2 + y 2 - 3 x + 2 y + k 4 = 0 Centre 3 2 , - 1 ; r = 13 - k 2 ⇒ k ≤ 13         … 1 case (i) Point 1 , - 1 3 lies on or inside circle C ⇒ S 1 ≤ 0 ⇒ k ≤ 92 9 ⋯ 2 case (ii) C lies in 4 th quadrant r < 1 ⇒ 13 - k 2 < 1 ⇒ k < 9         ⋯ 3 Hence from equation 1 ∩ 2 ∩ 3 ⇒ k ∈ ( 9 , 92 9 ]