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 $|\overrightarrow{x}|\le \sqrt{37},$ then $\lambda$ belongs to

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

Answer 1

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

Using section formula
$\overrightarrow{x}=\frac{\lambda \left(5\hat{j}\right)+5\hat{i}}{\lambda +1}$
$|\overrightarrow{x}|\le \sqrt{37}$
$\Rightarrow 25({\lambda }^2+1)\le 37(\lambda +1{)}^2$
$\Rightarrow 12{\lambda }^2+74\lambda +12\ge 0$
$\Rightarrow (6\lambda +1)(\lambda +6)\ge 0$
$\Rightarrow (-\infty ,-6]\cup [-\frac{1}{6},\infty )$