Question

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 $\left(1,-\frac{1}{3}\right)$ lies on or inside the circle $C$ is :

• A. An empty set
• B. $\left(6, \frac{95}{9}\right]$
• C. $\left[\frac{80}{9}, 10\right)$
• D. $\left(9, \frac{92}{9}\right]$

Answer 1

Answer: (D)

$C: 4 x^{2}+4 y^{2}-12 x+8 y+k=0$
$\Rightarrow x^{2}+y^{2}-3 x+2 y+\left(\frac{k}{4}\right)=0$
Centre $\left(\frac{3}{2},-1\right) ; r=\sqrt{\frac{13-k}{2}}$
$\Rightarrow k \leq 13 \ldots$ (1)
(i) Point $\left(1, \frac{-1}{3}\right)$ lies on or inside circle $C$
$\Rightarrow S _{1} \leq 0 \Rightarrow k \leq \frac{92}{9}$ ...(2)
(ii) $C$ lies in $4^{\text {th }}$ quadrant

$r <1$
$\Rightarrow \frac{\sqrt{13- k }}{2}<1$
$\Rightarrow k <9 \ldots .(3)$
Hence $(1) \cap(2) \cap(3) \Rightarrow k \in\left(9, \frac{92}{9}\right]$