If $\vec{b}$ is the vector whose initial point divides the joining of $5 \hat{i}$ and $5 \hat{j}$ in the ratio k : 1 and terminal point is origin. Also $|\vec{b}| \leq \sqrt{37}$ then the interval in which k lies 

$(-\infty,-6] \cup[-1 / 6, \infty)$

The point that divides $5 \hat{i}$ and $5 \hat{j}$ in the ratio of k : 1 is given by $\vec{b}=\frac{5 \hat{i}+5 k \hat{j}}{k+1}\left\{\right.$as terinal position is origin and initial is $\left.\frac{(5 \hat{j}) k+5 \hat{i}}{k+1}\right\}$

Also $|\vec{b}| \leq \sqrt{37} \Rightarrow \frac{1}{K+1} \sqrt{25+25 k^2} \leq \sqrt{37} \Rightarrow 5 \sqrt{1+k^2} \leq \sqrt{37}(k+1)$

On squaring both sides, we get $25\left(1+k^2\right) \leq 37\left(k^2+2 k+1\right)$

Or $12 k^2+74 k+12 \geq 0 \Rightarrow(6 k+1)(k+6) \geq 0$

Hence $k \in(-\infty,-6] \cup[-1 / 6, \infty)$

Hence (1) is correct answer.