Question

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.

Answer 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)`

Answer 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(x₁, y₁) and B(x₂, y₂) 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)`