question_answer

In what ratio does the point \[\left( \frac{24}{11},y \right)\] divide the line segment joining the points \[P(2,-2)\] and \[Q(3,7)\]? Also find the value of y.

Answer:

Let point R divides PQ in the ratio \[k:1\]

\[R=\left( \frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}},\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right)\]

\[\Rightarrow \]    \[\left( \frac{24}{11},y \right)=\left( \frac{k(3)+1(2)}{k+1},\frac{k(7)+1(-2)}{k+1} \right)\]

\[=\left( \frac{3k+2}{k+1},\frac{7k-2}{k+1} \right)\]

\[\Rightarrow \]            \[\frac{3k+2}{k+1}=\frac{24}{11}\]

\[\Rightarrow \]        \[11(3k+2)=24(k+1)\]

\[\Rightarrow \]         \[33k+22=24k+24\]

\[\Rightarrow \]       \[33k-24k=24-22\]

\[\Rightarrow \]                 \[9k=2\Rightarrow k=2/9\]

\[\therefore \]                   \[k:1=2:9\]

Now,                 \[y=\frac{7k-2}{k+1}=\frac{7\left( \frac{2}{9} \right)-2}{\frac{2}{9}+1}\]

\[=\frac{\frac{14}{9}-2}{\frac{2}{9}+1}=\frac{\frac{14-18}{9}}{\frac{2+9}{9}}=\frac{-4}{11}\]

Line PQ divides in the ratio \[2:9\] and value of \[y=\frac{-4}{11}\]