Question

Determine the ratio in which the line y = 2 + 3x divides the line segment AB joining the points A(–3, 9) and B(4, 2).

Answer 1

Given: Line: y = 2 + 3x. Points: A(–3, 9), B(4, 2).

Step-wise calculation:

1. Use a parametric/section approach:

Let P be the intersection point

P = A + t(B – A) 

= (–3 + 7t, 9 – 7t) 

So, AP : PB = t : (1 – t).

2. P lies on y = 2 + 3x.

So, substitute: 9 – 7t = 2 + 3(–3 + 7t)

9 – 7t = 2 – 9 + 21t

9 – 7t = –7 + 21t 

16 = 28t

⇒ t = `16/28`

⇒ t = `4/7`

3. Coordinates of P:

`x = -3 + 7(4/7)`

= 1

`y = 9 - 7(4/7)`

= 5

4. Ratio AP : PB = t : (1 – t)

= `4/7 : 3/7`   ...(Internal division)

= 4 : 3 

The line y = 2 + 3x divides segment AB internally in the ratio 4 : 3 at the point P(1, 5).