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Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

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Here, given points are A (-4, 3) and B ( 2, 8) and let the point dividing the line joining two points be P(m,6). 

Let the ratio be m:n

By section formula,

x = \(\frac{mx_2+nx_1}{m+n}\), y = \(\frac{my_2+ny_1}{m+n}\)

For point P(m,6),

m = \(\frac{m\times2+n\times(-4)}{m+n}\) …..(1)

And 6 = \(\frac{m\times8+n\times3}{m+n}\)…..(2)

Solving 2 for finding ratio between m and n,

 6 = \(\frac{m\times8+n\times3}{m+n}\)

6(m + n) = 8m +3n 

6m + 6n = 8m +3n 

∴ 2m = 3n

∴ \(\frac{m}n\) = \(\frac{3}2\)

∴ m : n = 3 : 2 

Now solving for equation 1, where m = 3 and n =2

m = \(\frac{m\times2+n\times(-4)}{m+n}\)

∴ m = \(\frac{6-8}5\)

∴ m = \(\frac{-2}5\)

Hence, our point is (\(\frac{-2}5\),6)

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