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Mathematics

In what ratio does the point (1, a) divide the join of (-1, 4) and (4, -1) ?

Also, find the value of a.

Section Formula

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Answer

We know that,

x=m1x2+m2x1m1+m2x = \dfrac{m1x2 + m2x1}{m1 + m2}

Substituting values we get,

1=m1×4+m2×1m1+m2m1+m2=4m1m2m2+m2=4m1m12m2=3m1m1m2=23.\Rightarrow 1 = \dfrac{m1 \times 4 + m2 \times -1}{m1 + m2} \\[1em] \Rightarrow m1 + m2 = 4m1 - m2 \\[1em] \Rightarrow m2 + m2 = 4m1 - m1 \\[1em] \Rightarrow 2m2 = 3m1 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{2}{3}.

∴ m1 : m2 = 2 : 3.

We know that,

y=m1y2+m2y1m1+m2y = \dfrac{m1y2 + m2y1}{m1 + m2}

Substituting values we get,

a=2×1+3×42+3a=2+125a=105a=2.\Rightarrow a = \dfrac{2 \times -1 + 3 \times 4}{2 + 3} \\[1em] \Rightarrow a = \dfrac{-2 + 12}{5} \\[1em] \Rightarrow a = \dfrac{10}{5} \\[1em] \Rightarrow a = 2.

Hence, ratio = 2 : 3 and a = 2.

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