Q) If A and B are (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = AB and P lies on the line segment AB.

Ans: 

Let the coordinates of P are (X, Y)

Since P lies on the line AP,

∴ AP + PB = AB … (1)

Given that AP = AB

∴ 7 AP = 3 AB …. (2)

Substituting value of AB from equation (1) , we get:

7 AP = 3 (AP + PB)

∴ 7 AP = 3 AP + 3 PB

∴ 4 AP =  3 PB

∴      … (3)

(m₁ and m₂ are the factors of division of the line)

Now, by section formula, coordinates of point P (X, Y) which lies between two points (x₁, y₁), (x₂, y₂) will be given by:

P (X,Y) =

Coordinates of the line segment AB are: (-2, -2) and (2, -4)

and from equation (3), we have values of m₁ and m₂

By substituting these values in above section formula, we get:

P (X,Y) =

P (X,Y) =

P (X,Y) =

Therefore, the coordinates of point P are