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 \[y=\frac{-4}{11}\]
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