Answer:
Here \[P(x,y)\] divides line segment AB such that \[AP=\frac{3}{7}AB\] \[\Rightarrow \frac{AP}{AB}=\frac{3}{7}\] \[\Rightarrow \frac{AP}{AB}=\frac{7}{3}\] \[\Rightarrow \frac{AB}{AP}-1=\frac{7}{3}-1\] \[\Rightarrow \frac{AB-AP}{AP}=\frac{4}{3}\] \[\Rightarrow \frac{BP}{AP}=\frac{4}{3}\] \[\Rightarrow \frac{AP}{BP}=\frac{3}{4}\] \[\therefore \] P divides AB in the ratio \[3:4\text{ (}m:n)\] The coordinates of P are \[(x,y)\] Therefore, \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\] \[x=\frac{3\times 2+4(-2)}{3+4},y=\frac{3(-4)+4(-2)}{3+4}\] \[x=\frac{6-8}{7},y=\frac{-12-8}{7}\] \[x=\frac{-2}{7},y=\frac{-20}{7}\] Therefore, co-ordinates of \[P(x,y)\] are \[\left( \frac{-2}{7},\frac{-20}{7} \right)\]
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