Question

If the point $(2,k)$ lies outside the circle's $x^2+y^2+x-2y-14=0$ and $x^2+y^2=13$ then range of $k$ is

• A. $(-\infty ,-2)\cup (4,\infty )$
• B. $(-\infty ,-3)\cup (4,\infty )$
• C. $(-\infty ,-3)\cup (3,\infty )$
• D. $(-\infty ,-1)\cup (4,\infty )$

Answer 1

The correct option is B $(-\infty ,-3)\cup (4,\infty )$

For an external point to circle $S>0$
so,
$4+k^2+2-2k-14>0$
$\Rightarrow k^2-2k-8>0$
$\Rightarrow (k-4)(k+2)>0$
$\Rightarrow k\in (-\infty ,-2)\cup (4,\infty )\cdots \left(1\right)$
For $2^{\text{nd}}$ circle,
$\Rightarrow 4+k^2-13>0$
$\Rightarrow k^2-9>0$
$\Rightarrow k\in (-\infty ,-3)\cup (3,\infty )\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, we get
$k\in (-\infty ,-3)\cup (4,\infty )$