If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that `AP = 3/7 AB` and P lies on the line segment AB.

#### Solution 1

The coordinates of point A and B are (−2, −2) and (2, −4) respectively.

Since AP = `3/7 AB`

Therefore, AP: PB = 3:4

Point P divides the line segment AB in the ratio 3:4.

Coordinates of P = `((3xx2+4xx(-2))/(3+4), (3xx(-4)+4xx(-2))/(3+4))`

`= ((6-8)/7, (-12-8)/7)`

`=(-2/7, -20/7)`

#### Solution 2

We have two points A (-2,-2) and B (2,-4). Let P be any point which divides AB as

`AP = 3/7 AB`

Since,

AB = (AP + BP)

So,

7AP = 3AB

7AP = 3(AP + BP)

4AP = 3BP

`(AP)/(BP) = 3/4`

Now according to the section formula if any point P divides a line segment joining `A(x_1, y_1)` and `B(x_2, y_2)` in the ratio m: n internally than,

`P(x,y) = ((nx_1 + mx_2)/(m + n)"," (ny_1 + my_2)/(m + n))`

Therefore P divides AB in the ratio 3: 4. So,

`P(x,y) = ((3(2) + 4(-2))/(3 + 4)"," (3(-4) + 4(-2))/(3 + 4))`

`= (-2/7,-20/7)`