If $b$ is a vector whose initial point divides the join of $5\hat{i}$ and $5\hat{j}$ in the ratio $k:1$ and whose terminal point in the origin and $\left|b\right|\le \sqrt{37}$, then$k$ lies in the interval

The point the divides $5\hat{i}$ and $5\hat{j}$ in the ratio of $k:1$ is $\frac{5\left(\hat{j}\right)k+5\left(\hat{i}\right)1}{k+1}$

$\therefore b= \frac{5\widehat{i }+5k\hat{j}}{k+1}$

Also,   $\left|b\right|\le \sqrt{37}$

$\Rightarrow \frac{1}{k+1}\sqrt{25+25k^2}\le \sqrt{37}$

or $5\sqrt{1+k^2}\le \sqrt{37}\left(k+1\right)$

On Squaring both sides, we get
$25(1+k^2) \le 37 (k^2+2K+1)$

or $6k^2+ 37k+6\ge 0$or $\left(6k+1\right)\left(k+6\right)\ge 0$

$k\in \left(-\infty ,-6\right)\cup \left[-\frac{1}{6},\infty \right]$