question_answer

The coordinates of the points A, B, C are \[({{x}_{1}},{{y}_{1}})\], \[({{x}_{2}},{{y}_{2}})\], \[({{x}_{3}},\,{{y}_{3}})\] and D divides the line AB in the ratio l : k. If P divides the line DC in the ratio m : k + l, then the coordinates of P are

A) \[\left( \frac{k{{x}_{1}}+l{{x}_{2}}+m{{x}_{3}}}{k+l+m},\,\frac{k{{y}_{1}}+l{{y}_{2}}+m{{y}_{3}}}{k+l+m} \right)\]

B) \[\left( \frac{l{{x}_{1}}+m{{x}_{2}}+k{{x}_{3}}}{l+m+k},\,\frac{l{{y}_{1}}+m{{y}_{2}}+k{{y}_{3}}}{l+m+k} \right)\]

C) \[\left( \frac{m{{x}_{1}}+k{{x}_{2}}+l{{x}_{3}}}{m+k+l},\,\frac{m{{y}_{1}}+k{{y}_{2}}+l{{y}_{3}}}{m+k+l} \right)\]

D) None of these

Correct Answer: A

Solution :

Coordinates of D will be \[\left( \frac{l{{x}_{2}}+k{{x}_{1}}}{l+k},\,\,\frac{l{{y}_{2}}+k{{y}_{1}}}{l+k} \right)\] Now again, DC is divided by P in \[m:k+l.\] Then the coordinates of P will be given by \[\left( \frac{m{{x}_{3}}+l{{x}_{2}}+k{{x}_{1}}}{k+l+m},\,\frac{m{{y}_{3}}+l{{y}_{2}}+k{{y}_{1}}}{k+l+m} \right)\].