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Determine the ratio in which the line $2x + y = 6$ divides the line segment joining the points $(1, 3)$ and $(2, 5)$.
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Let the line divides the line segment in the ratio $k : 1$ at P (1/2 Mark)
Coordinates of P are $(\frac{2k+1}{k+1}, \frac{5k+3}{k+1})$ (1 Mark)
P lies on $2x + y = 6$
$2(\frac{2k+1}{k+1}) + (\frac{5k+3}{k+1}) = 6$ (1/2 Mark)
$2(2k + 1) + (5k + 3) = 6(k + 1)$
$4k + 2 + 5k + 3 = 6k + 6$
$9k + 5 = 6k + 6$
$3k = 1 \Rightarrow k = \frac{1}{3}$ (1 Mark)
Required ratio is $1:3$
Coordinates of P are $(\frac{2k+1}{k+1}, \frac{5k+3}{k+1})$ (1 Mark)
P lies on $2x + y = 6$
$2(\frac{2k+1}{k+1}) + (\frac{5k+3}{k+1}) = 6$ (1/2 Mark)
$2(2k + 1) + (5k + 3) = 6(k + 1)$
$4k + 2 + 5k + 3 = 6k + 6$
$9k + 5 = 6k + 6$
$3k = 1 \Rightarrow k = \frac{1}{3}$ (1 Mark)
Required ratio is $1:3$