question_answer

Find the ratio in which the line segment joining the points \[(-3,10)\]and is divided by \[(-1\,\,,6).\]

A)  2 : 7    

B)                     3 : 5

C)  4 : 1                            

D)  3 : 7

Correct Answer: A

Solution :

Let the point A \[(-1,6)\] divide the line joining B\[(-\,\,3,10)\] and C\[(6,-\,\,8)\] in the ratio k : 1. Then. the coordinates of A are \[\left( \frac{6k-3}{k+1},\frac{-8k+10}{k+1} \right).\]

\[\left[ \because \text{Internally}\,\,\text{ratio}\left( \frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}},\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right) \right]\]

But, the co-ordinates of A are given by \[(-1,6)\]

On comparing coordinates, we get

\[\frac{6k-3}{k+1}=-\,\,1\] and \[\frac{-\,\,8k+1}{k+1}=6\]

\[\Rightarrow \]   \[6k-3=-k-1\]and \[-\,\,8k+10=6k+6\]

\[\Rightarrow \]   \[6k+k=-1+3\] and \[-\,\,8k+6k=6-10\]

\[\Rightarrow \]   \[7k=2\]and\[-14k=-\,\,4\therefore k=\frac{2}{7}\]

So, the point A divides BC in the ratio of 2 : 7.