question_answer

Find the ratio in which the point \[P\left( \frac{3}{4},\frac{5}{12} \right)\] divides the line segment joining the points \[A\left( \frac{1}{2},\frac{3}{2} \right)\] and \[B(2,-5)\].

Answer:

Let point \[P\left( \frac{3}{4},\frac{5}{12} \right)\] divides the line AB in ratio \[k:1\]

Then, by section formula, coordinates of P are

\[\frac{2k+\frac{1}{2}}{k+1}=\frac{3}{4}\]

and                   \[\frac{-5k+\frac{3}{2}}{k+1}=\frac{5}{12}\]

\[\Rightarrow 8k+2=3k+3\]

and                   \[-60k+18=5k+5\]

\[\Rightarrow 8k-3k=3-2\]

and                   \[65k=18-5\]

\[\Rightarrow 5k=1\]

and                   \[65k=13\Rightarrow k=\frac{1}{5}\]in each case

Hence the required ratio is \[1:5\].