In what ratio is the line segment joining the pair of points 2 4 and 3 6 is divided by x axis

Question 5 Coordinated Geometry - Exercise 7.3

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Answer:

Let the ratio in which x-axis divides the line segment joining (–4, –6) and (–1, 7) = 1: k.

Then,

x-coordinate becomes, \frac{\left(-1-4k\right)}{(k+1)}

y-coordinate becomes, \frac{\left(7-6k\right)}{(k+1)}

Since P lies on x-axis, y coordinate = 0

\frac{\left(7-6k\right)}{(k+1)}=0\\ 7-6k=0\\ k=\frac{7}{6}

Therefore, the point of division divides the line segment in the ratio 6 : 7.

Now, m1 = 6 and m2 = 7

By using the section formula,

x=\frac{\left(m_1x_2+m_2x_2\right)}{(m_1+m_2)}=\frac{\left[6(-1)+7(-4)\right]}{(6+7)}=\frac{\left(-6-28\right)}{13}=-\frac{34}{13}\\ So,\ now\\ y=\frac{\left[6(7)+7(-6)\right]}{(6+7)}=\frac{\left(42-42\right)}{13}=0

Hence, the coordinates of P are (-34/13, 0)

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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 2

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 3

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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