Determine the ratio in which the line 2x + y -4 = 0, Mathematics

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Determine the ratio in which the line 2x + y -4 = 0 divide the line segment joining the points A (2,-2) and B (3, 7).Also find the coordinates of the point of division.

[Ans:2 : 9 , ( 24 ,- 4 )]

Ans : Let the ratio be k:1

Let the co-ordinates of point of division be (x, y)

∴ x = k (3) +1.2/k +1 = 3k + 2/k +1

y =k (7) -1.2/ k + 1 = = 7k - 2/k +1

(x, y) lies on the line 2x + y - 4 = 0.

∴ 2( 3k + 2/k +1)+( 7k - 2 /k+1)- 4 = 0

2(3k+2) + (7k-2) - 4 (k+1) = 0

6k + 4 + 7k - 2 - 4k - 4 = 0

9k - 2 = 0 k = 2/9

Ratio is 2:9

∴ x (3x 2/9 + 2)/(2/9+1) = (2/3 + 2)/(11/9) = (2 + 6/3)/(11/9) = 8/3 x 9/11 = 24/11

y = (7(2/9)-2)2/9+1 = (14 -18/9)/(11/9) = -4/9 x 9/11 = - 4/11

∴(x, y) =( 24/11 , - 4/11 )

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