Answer:
Given, \[A(1,2)\] and \[B(6,7)\] are the given points of a line segment AB with a point P on it. Let the co-ordinate of point P be \[(x,y)\] \[AP=\frac{2}{3}AB\] (Given) \[AB=AP+PB\] \[\Rightarrow \frac{AP}{PB}=\frac{2}{3}\] \[\therefore m=2,n=3\] Then, by section formula, we have \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\] \[x=\frac{2\times 6+3\times 1}{2+3}\] and \[y=\frac{2\times 7+3\times 2}{2+3}\] \[x=\frac{15}{2}\] and \[y=\frac{20}{5}\] \[\therefore \,\,x=3\] and \[\,y=4\] Hence, the required point is \[P(3,4)\].
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