Answer:
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), \[\frac{-3(1)+3}{1+1}=m\] \[\Rightarrow \] \[\frac{-3+3}{2}=m\] \[\Rightarrow \] \[m=0\]
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