The set of values of k for which the circle C: 4 x ^ 2 + 4 y ^ 2 -...

Given equation of circle $C: 4{x}^{2} + 4{y}^{2} - 12x + 8y + k = 0$

$⇒ {x}^{2} + {y}^{2} - 3x + 2y + \left(\dfrac{k}{4}\right) = 0$

Centre \left(\dfrac{3}{2}, - 1\right); r = \sqrt{\dfrac{13 - k}{2}} ⇒ k \le 13 \ldots $\left(1\right)$

case (i) Point \left(1, \dfrac{- 1}{3}\right)$lies on or inside circle$C$</p><p>$⇒ {S}_{1} \le 0 ⇒ k \le \dfrac{92}{9} ⋯ \left(2\right)$</p><p>case (ii)$C$lies in${4}^{th}$quadrant</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/88e4e686-abef-419a-83a3-e364eaba269f-image.png" style="width: 140px; height: 101px;" /></p><p>$r < 1$</p><p>$⇒ \dfrac{\sqrt{13 - k}}{2} < 1$</p><p>$⇒ k < 9 ⋯ \left(3\right)$</p><p>Hence from equation \left(1\right) \cap \left(2\right) \cap \left(3\right) ⇒ k \in ( 9, \dfrac{92}{9} ]$