A) 1 : 2
B) 2 : 1
C) 2 : 3
D) 3 : 4
Correct Answer: C
Solution :
Given, equation of line \[y-x+2=0\] and co-ordinates of points \[({{x}_{1}},\,{{y}_{1}})=(3,\,-1)\] and \[({{x}_{2}},\,{{y}_{2}})=(8,\,9)\]. We know that if the ratio in which a line \[ax+by+c=0\] is divided by points \[({{x}_{1}},\,{{y}_{1}})\] and \[({{x}_{2}},\,{{y}_{2}})\] is \[\lambda :1\], then intersecting point \[\left( \frac{\lambda {{x}_{2}}+{{x}_{1}}}{\lambda +1},\,\frac{\lambda {{y}_{2}}+{{y}_{1}}}{\lambda +1} \right)\] lies on \[ax+by+c=0\]. Thus any point on the line joining \[(3,\,-1)\] and \[(8,\,9)\] dividing it in the ratio \[\lambda :1\] is \[\left( \frac{8\lambda +3}{\lambda +1},\,\frac{9\lambda -1}{\lambda +1} \right)\] and if it lies on \[y-x+2=0,\] then \[\frac{9\lambda -1}{\lambda +1}-\frac{8\lambda +3}{\lambda +1}+2=0\] or \[9\lambda -1-(8\lambda +3)+2(\lambda +1)=0\] or \[3\lambda -2=0,\,\lambda =\frac{2}{3}\] i.e. ratio is \[2:3\].
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