Question

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

Answer 1

Given: A(–2, –2), B(2, –4). P lies on AB with AP = `3/7` AB.

Step-wise calculation:

1. `(AP)/(AB) = 3/7`

⇒ `(PB)/(AB) = 4/7`

⇒ AP : PB = 3 : 4

2. Using the section formula:

If P divides AB internally in the ratio m₁ : m₂ (with AP : PB = m₁ : m₂),

Then `x = (m_1xx x_2 + m_2 xx x_1)/(m_1 + m_2)` 

`y = (m_1 xx y_2 + m_2 xx y_1)/(m_1 + m_2)`

3. Here m₁ = 3, m₂ = 4

x₁ = –2, y₁ = –2

x₂ = 2, y₂ = −4 

`x = (3 xx 2 + 4 xx (-2))/(3 + 4)` 

= `(6 - 8)/7` 

= `(-2)/7`

`y = (3 xx (-4) + 4 xx (-2))/(3 + 4)` 

= `(-12 - 8)/7`

= `(-20)/7`

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