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 $|\bar{a}| \leq 2$, then find how many integer values of $m$ lie in the interval $[-2,2]$.

Answer 1

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