Question

If $\bar{a}$ is the vector whose initial point divides the joining of $2\hat{i}$ and $2\hat{j}$ in the ratio $m:1$ and terminal point is origin. Also $|\bar{a}|\le 2$, then find how many integer values of $m$ lie in the interval $[-2,2]$.

Answer 1

Answer: 3

Here,
$\bar{a}=\frac{m(2\hat{j})+1(2\hat{i})}{m+1}$
$\Rightarrow \bar{a}=\frac{2\hat{i}+2m\hat{j}}{m+1}$
$|\bar{a}|=\frac{1}{m+1}\sqrt{4+4m^2}\ge 0$
$\Rightarrow m+1>0$
$\Rightarrow$ Feasible region $=(-1,\infty )...(i)$
Also, $|\bar{a}|\le 2$
$\Rightarrow \frac{\sqrt{4+4m^2}}{m+1}\le 2$
$\Rightarrow \sqrt{m^2+1}\le m+1$
$\Rightarrow m^2+1\le m^2+1+2m$
$\dots$ [By squaring both the sides]
$\Rightarrow 2m\ge 0$
$\Rightarrow m\ge 0...(ii)$
$\Rightarrow m\in [0,\infty )\dots [$ From (i) and (ii) $]$
$\Rightarrow$ Number of integer values of $m$ lying in $[-2,2]$ is $3.$