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.

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), \[\frac{-3(1)+3}{1+1}=m\]

\[\Rightarrow \]   \[\frac{-3+3}{2}=m\]

\[\Rightarrow \]   \[m=0\]