question_answer

Find the coordinates of a point P on the line segment joining \[A(1,2)\] and \[B(6,7)\] such that\[AP=\frac{2}{5}AB\].

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 \[P(3,4)\].