• # question_answer Find the ratio in which $P\,(4,m)$ divides the line segment joining the points $A\text{ (}2,3)$ and$B\text{ (}6,-3)$. Hence find m.

 Let P divides line segment AB in the ratio $k:1$. Coordinates of P $P=\left( \frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}},\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right)$ $(4,m)=\left( \frac{k\times 6+1\times 2}{k+1},\frac{k\times (-3)+1\times 3}{k+1} \right)$ $(4,m)=\left( \frac{6k+2}{k+1},\frac{-3k+3}{k+1} \right)$ On comparing, we get $\left( \frac{6k+2}{k+1} \right)=4$ $\Rightarrow$   $6k+2=4+4k$ $\Rightarrow$   $6k-4k=4-2$ $\Rightarrow$           $2k=2$ $\Rightarrow$             $k=1$ Hence, P divides AB in the $1:1$. From (i), $\frac{-3(1)+3}{1+1}=m$ $\Rightarrow$   $\frac{-3+3}{2}=m$ $\Rightarrow$   $m=0$