Question

If $\overrightarrow{b}$ is the vector whose initial point divides the joining $5\overrightarrow{i}$ and $5\overrightarrow{j}$ in the ratio $\lambda :1$ and terminal point is at origin. If $|\overrightarrow{b}|\le \sqrt{37},$ then set of values of $\lambda$ is

• A. $(-\infty ,-6]\cup [-\frac{1}{6},\infty )$
• B. $(-6,6)$
• C. $(-\infty ,-2)\cup (2,\infty )$
• D. $[-2,2]$

Answer 1

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

Let point $B$ divides the vector $5\overrightarrow{i}$ and $5\overrightarrow{j}$ in $\lambda :1.$
So, $\overrightarrow{OB}=\overrightarrow{b}=\frac{5\overrightarrow{i}+5\lambda \overrightarrow{j}}{\lambda +1}$
and $|\overrightarrow{b}|\le \sqrt{37}$
$\Rightarrow \frac{25+25{\lambda }^2}{(\lambda +1{)}^2}\le 37\Rightarrow 6{\lambda }^2+37\lambda +6\ge 0\Rightarrow (6\lambda +1)(\lambda +6)\ge 0$
$\Rightarrow \lambda \in (-\infty ,-6]\cup [-\frac{1}{6},\infty )$