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. |
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