Question

The set of values of $k$ for which the circle $C:4x^2+4y^2-12x+8y+k=0$ 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)

$C:4x^2+4y^2-12x+8y+k=0$
$\Rightarrow x^2+y^2-3x+2y+\left(\frac{k}{4}\right)=0$
Centre $\left(\frac{3}{2},-1\right);r=\sqrt{\frac{13-k}{2}}$
$\Rightarrow k\le 13\dots$ (1)
(i) Point $\left(1,\frac{-1}{3}\right)$ lies on or inside circle $C$
$\Rightarrow S_1\le 0\Rightarrow k\le \frac{92}{9}$ ...(2)
(ii) $C$ lies in $4^{\text{th }}$ quadrant

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