If the roots of $x^2-2kx+k^2-1=0$ lies between $-2$ and $4$then the interval in which $k$ lies, is

Given that both roots of the equation $f\left(x\right)\equiv x^2-2kx+k^2-1=0$ lies in between $-2$ and $4$

It implies that $b^2-4ac>0$ and $f\left(-2\right)>0,f\left(4\right)>0$ since $a=1>0$

Hence,

$\begin{array}{rcl}f\left(-2\right)& >& 0\\ 4+4k+k^2-1& >& 0\\ k^2+4k+3& >& 0\\ \left(k+1\right)\left(k+4\right)& >& 0\end{array}$

and 

$\begin{array}{rcl}f\left(4\right)& >& 0\\ 16-8k+k^2-1& >& 0\\ k^2-8k+15& >& 0\\ \left(k-5\right)\left(k-3\right)& >& 0\end{array}$

It implies that 

$\begin{array}{rcl}k& \in & \left(-\infty ,-4\right)\cup \left(-1,\infty \right)\cap \left(-\infty ,3\right)\cup \left(5,\infty \right)\\ & \in & \left(-1,3\right)\cup \left(5,\infty \right)\end{array}$

Therefore, among the given options, $\left(-1,3\right)$ is right answer.