Question

A line AB meets X-axis at A and Y-axis at B. P(4, –1) divides AB in the ration 1 : 2.

1. Find the co-ordinates of A and B.
2. Find the equation of the line through P and perpendicular to AB.

Answer 1

i. Since A lies on the X-axis, let the co-ordinates of A be (x, 0).

Since B lies on the Y-axis, let the co-ordinates of B be (0, y).

Let m = 1 and n = 2

Using section formula,

Coordinates of P = `((1(0) + 2(x))/(1 + 2), (1y + 2(0))/(1 + 2))`

`=> (4, -1) = ((2x)/3, y/3)`

`=> (2x)/3 = 4` and `y/3 = -1`

`=>` x = 6 and y = –3

So, the co-ordinates of A are (6, 0) and that of B are (0, –3). 

ii. Slope of AB = `(-3 - 0)/(0 - 6) = (-3)/(-6) = 1/2`  

`=>` Slope of line perpendicular to AB = m = –2

P = (4, –1)

Thus, the required equation is

y – y₁ = m(x – x₁)

`=>` y – (–1) = –2(x – 4)

`=>` y + 1 = –2x + 8

`=>` 2x + y = 7