10th Class Mathematics Solved Paper - Mathematics-2018

  • question_answer Find the ratio in which \[P\,(4,m)\] divides the line segment joining the points \[A\text{ (}2,3)\] and\[B\text{ (}6,-3)\]. Hence find m.

    Answer:

    Let P divides line segment AB in the ratio \[k:1\].
    Coordinates of P
                         \[P=\left( \frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}},\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right)\]
                    \[(4,m)=\left( \frac{k\times 6+1\times 2}{k+1},\frac{k\times (-3)+1\times 3}{k+1} \right)\]
                    \[(4,m)=\left( \frac{6k+2}{k+1},\frac{-3k+3}{k+1} \right)\]
    On comparing, we get
                \[\left( \frac{6k+2}{k+1} \right)=4\]
    \[\Rightarrow \]   \[6k+2=4+4k\]
    \[\Rightarrow \]   \[6k-4k=4-2\]
    \[\Rightarrow \]           \[2k=2\]
    \[\Rightarrow \]             \[k=1\]
    Hence, P divides AB in the \[1:1\].
    From (i), \[\frac{-3(1)+3}{1+1}=m\]
    \[\Rightarrow \]   \[\frac{-3+3}{2}=m\]
    \[\Rightarrow \]   \[m=0\]


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