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. None of these

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 of $k:1$ is
$=\frac{\left(5\hat{j}\right)k+\left(5\hat{i}\right)1}{k+1}$
$\therefore \overrightarrow{b}=-\left(\frac{5\hat{i}+5k\hat{j}}{k+1}\right)$

Also, $|\overrightarrow{b}|\le \sqrt{37}$
$\Rightarrow \frac{1}{k+1}\sqrt{25+25k^2}\le \sqrt{37}$
$\Rightarrow 5\sqrt{1+k^2}\le \sqrt{37}(k+1)$
Squaring both sides, we get
$25(1+k^2)\le 37(k^2+2k+1)$
or, $6k^2+37k+6\ge 0$
or, $(6k+1)(k+6)\ge 0$
or, $k\in (-\infty ,-6]\cup [-\frac{1}{6},\infty )$