In what ratio does the x-axis divide the line segment joining the points (2, –3) and (5, 6)?

In what ratio is the line joining (2, -3) and (5, 6) divided by the x-axis.

Let the line joining points A (2, −3) and B (5, 6) be divided by point P (x, 0) in the ratio k : 1.

`y=(ky_2+y_1)/(k+1)`

`0=(kxx6+1xx(-3))/(k+1)`

`0=6k-3`

`k=1/2`

Thus, the required ratio is 1: 2.

Concept: Co-ordinates Expressed as (x,y)

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Page 2

In what ratio is the line joining (2, -4) and (-3, 6) divided by the y – axis.

Let the line joining points A (2, -4) and B (-3, 6) be divided by point P (0, y) in the ratio k : 1.

`x=(kx_2+x_1)/(k+1)`

`0=(kxx(-3)+1xx2)/(k+1)`

`0=-3k+2`

`k=2/3`

Thus, the required ratio is 2: 3.

Concept: Co-ordinates Expressed as (x,y)

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Page 3

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

Let the point P (1, a) divides the line segment AB in the ratio k: 1. 
Using section formula, we have:

`1=(4k-1)/(k+1)`

`=>k+1=4k-1`

`=>2=3k`

`=>k=2/3`  ............(1)

`=>a=(-k+4)/(k+1)`

`=> a = (-2/3 + 4)/(2/3 + 1)`     (from 1)

`=> a = 10/5 = 2`

Hence, the required is 2 : 3 and the value of a is 2.

Concept: Co-ordinates Expressed as (x,y)

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Page 4

In what ratio does the point (a, 6) divide the join of (-4, 3) and (2, 8)? Also, find the value of a.

Let the point P (a, 6) divides the line segment joining A (-4, 3) and B (2, 8) in the ratio k: 1.
Using section formula, we have:

`6=(8k+3)/(k+1)`

`=> 6k+6=8k+3`

`=>3=2k`

`=>k=3/2`  .................(1)

`=>a=(2k-4)/(k+1)`

`=>a=(2xx3/2-4)/(3/2+1)`   (from equation 1)

`=>a=-2/5`

Hence, the required ratio is 3:2 and the value of a is `-2/5`

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