Question

If $\overrightarrow{x}$ is a vector whose initial point divides the line joining $5\hat{i}$, and $5\hat{j}$ in the ratio $\lambda :1$ and the terminal point is the origin. Also given $\left|\overrightarrow{x}\right|\le \sqrt{37}$, then $\lambda$ belongs to

• A. $[-\frac{1}{6},\frac{1}{6}]$
• B. $(-\infty ,-6)\cup (-\frac{1}{6},\infty )$
• C. $(-\infty ,-8)$
• D. $(1,\infty )$

Answer 1

The correct option is B $(-\infty ,-6)\cup (-\frac{1}{6},\infty )$

From the condition given for $\overrightarrow{x}$ we have

$\frac{|\lambda \left(5j\right)+5i|}{\lambda +1}\le \sqrt{37}$

$25({\lambda }^2+1)\le 37(\lambda +1{)}^2$

$\Rightarrow (6\lambda +1)(\lambda +6)\ge 0$

$\lambda \in (-\infty ,-6)\cup (-\frac{1}{6},\infty )$