Determine the ratio in which the line $2 x + y$ $=$ $4$ divides the line segment joining the points $(2, - 2)$ and $(3, 7)$

1 : 3

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2 : 9

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1 : 2

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3 : 4

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Solution

The correct option is B $2 : 9$
Let there be any point $P (x_{1}, y_{1})$ which is the common point of both lines and divides the given line $2 x + y = 4$ in ration $k : 1.$

we have,
$x_{1} = \frac{m x_{2} + n x_{1}}{m + n}$ , $y_{1} = \frac{m y_{2} + n y_{1}}{m + n}$ for $m : n$ ratio

$∴ x_{1} = \frac{3 k + 2}{k + 1}$ and $y_{1} = \frac{7 k - 2}{k + 1}$

this point lie on $2 x + y = 4$

$∴ 2 (\frac{3 k + 2}{k + 1}) + (\frac{7 k - 2}{k + 1}) = 4$

$\Rightarrow 6 k + 4 + 7 k - 2 = 4 k + 4$

$\Rightarrow 13 k + 2 = 4 k + 4 \Rightarrow 9 k = 2 \Rightarrow k = 2 / 9$

So line $2 x + y = 4$ divides the join of $(2, - 2)$ and $(3, 7)$ in ration $2 : 9.$

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