Question

If $\overrightarrow{b}$ 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 is the origin and $|\overrightarrow{b}|\le \sqrt{37}$, then $k$ lies in the interval

• A. $[-6,-\frac{1}{6}]$
• B. $(-\infty ,-6]\cup [-\frac{1}{6},\infty )$
• C. $(0,6)$
• D. $[0,6]$

Answer 1

The correct option is B

$(-\infty ,-6]\cup [-\frac{1}{6},\infty )$

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

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

Also, $|\overrightarrow{b}|\le \sqrt{37}$

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

$=5\sqrt{1+k^2}\le \sqrt{37}(k+1)$

squaring both the sides, we get

$25(1+k^2)\le 37(k^2+2k+1)$

$=6k^2+37k+6\le 0$

$=(6k+1)(k+6)\ge 0$

$\Rightarrow kϵ(-\infty ,-6]\cup [-\frac{1}{6},\infty )$