Question

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

1) Find the coordinates of A and B.

2) Find the equation of the line through P and perpendicular to AB.

Answer 1

1) Since A lies on the x-axis, let the coordinates of A be (x, 0).

Since B lies on the y-axis, let the coordinates 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 coordinates of A are (6,0) and that of B are (0, -3).

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

=> Slope of line perpendicular to AB=m= -2

P = (4,-1)

⇒Required equation is

`y - y_1 = m(x - x_1)`

`=> y - (-1) = -2(x  -  4)`

`=> y + 1 = -2x + 8`

`=> 2x + y = 7`