1. Write down the co-ordinates of the point P that divides the line joining `A(-4,1)` and `B(17,10)` in the ratio `1:2`

Solution:
Let `P(x,y)` divides the line segment joining the points AB in the ratio `1:2`.

The given points are `A(-4,1),B(17,10)`

`:. x_1=-4,y_1=1,x_2=17,y_2=10`

and `m:n=1:2`


By section formula
`x=(mx_2+nx_1)/(m+n)`

`=(1*17+2*-4)/(1+2)`

`=(17-8)/(3)`

`=(9)/(3)`

`=3`


`y=(my_2+ny_1)/(m+n)`

`=(1*10+2*1)/(1+2)`

`=(10+2)/(3)`

`=(12)/(3)`

`=4`


Hence, the co-ordinates of the point `P` are `(3,4)`

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2. Write down the co-ordinates of the point P that divides the line joining `A(5,12)` and `B(2,9)` in the ratio `2:1`

Solution:
Let `P(x,y)` divides the line segment joining the points AB in the ratio `2:1`.

The given points are `A(5,12),B(2,9)`

`:. x_1=5,y_1=12,x_2=2,y_2=9`

and `m:n=2:1`


By section formula
`x=(mx_2+nx_1)/(m+n)`

`=(2*2+1*5)/(2+1)`

`=(4+5)/(3)`

`=(9)/(3)`

`=3`


`y=(my_2+ny_1)/(m+n)`

`=(2*9+1*12)/(2+1)`

`=(18+12)/(3)`

`=(30)/(3)`

`=10`


Hence, the co-ordinates of the point `P` are `(3,10)`

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3. Write down the co-ordinates of the point P that divides the line joining `A(2,8)` and `B(6,14)` in the ratio `5:3` Externally

Solution:
Let `P(x,y)` divides the line segment joining the points AB in the ratio `5:3`.

The given points are `A(2,8),B(6,14)`

`:. x_1=2,y_1=8,x_2=6,y_2=14`

and `m:n=5:3`


By section formula
`x=(mx_2-nx_1)/(m-n)`

`=(5*6-3*2)/(5-3)`

`=(30-6)/(2)`

`=(24)/(2)`

`=12`


`y=(my_2-ny_1)/(m-n)`

`=(5*14-3*8)/(5-3)`

`=(70-24)/(2)`

`=(46)/(2)`

`=23`


Hence, the co-ordinates of the point `P` are `(12,23)`

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4. Write down the co-ordinates of the point P that divides the line joining `A(1,-3)` and `B(3,5)` in the ratio `5:3` Externally

Solution:
Let `P(x,y)` divides the line segment joining the points AB in the ratio `5:3`.

The given points are `A(1,-3),B(3,5)`

`:. x_1=1,y_1=-3,x_2=3,y_2=5`

and `m:n=5:3`


By section formula
`x=(mx_2-nx_1)/(m-n)`

`=(5*3-3*1)/(5-3)`

`=(15-3)/(2)`

`=(12)/(2)`

`=6`


`y=(my_2-ny_1)/(m-n)`

`=(5*5-3*-3)/(5-3)`

`=(25+9)/(2)`

`=(34)/(2)`

`=17`


Hence, the co-ordinates of the point `P` are `(6,17)`