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  • In what ratio does the x–axis divide the line segment joining the points (– 4, – 6) and (–1, 7) Find the coordinates of the point of division.

   In what ratio does the x–axis divide the line segment joining the points (– 4, – 6) and (–1, 7)? Find the coordinates of the point of division.

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Solution.
             
 \left ( X,Y \right )= \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 )

Let the point on x-axis (x, 0) when divides the given points (–4, –6) and (–1, 7) in the ration k : 1.
(x1, y1) =A (-4, -6)           (x2, y2) = B(-1, 7)
m1 = k, m2 = 1
Using section formula, we have
p\left ( x,0 \right )= \left [ \frac{-k-4}{k+1},\frac{7k-6}{k+1} \right ]
By comparing the left-hand side and the right-hand side we get
\frac{7k-6}{k+1}= 0
7k-6 = 0
k= \frac{6}{7}
x= \frac{-k-4}{k+1}= \frac{-\frac{6}{7}-4}{\frac{6}{7}+1}= \frac{-6-28}{6+7}= \frac{-34}{13}
The required ratio is 6: 7.
Co-ordinate \, of\, P\left ( \frac{-34}{13},0 \right )

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