Question

Prove that the point P dividing the line segment joining the points A(–1, 7) and B(4, –3) in the ratio 3 : 2, lies on the line x – 3y = –1. Also find length of PA and PB.

Answer 1

Find coordinates of P with m = 3, n = 2, A(–1, 7), B(4, –3):

Section Formula: `P(x, y) = ((mx_2 + nx_1)/(m + n), (my_2 + ny_1)/(m + n))`

`x = (3(4) + 2(-1))/(3 + 2)`

= `(12 - 2)/5`

= `10/5`

= 2

`y = (3(-3) + 2(7))/(3 + 2)`

= `(-9 + 14)/5`

= `5/5`

= 1

So, P is (2, 1).

Verification: Substitute P(2, 1) into x – 3y = –1:

L.H.S. = 2 – 3(1)

= 2 – 3

= –1

Since L.H.S. = R.H.S., point P lies on the line.

Calculating lengths:

Distance Formula: `d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

`PA = sqrt((2 - (-1))^2 + (1 - 7)^2`

= `sqrt(3^2 + (-6)^2`

= `sqrt(9 + 36)`

= `sqrt(45)`

= `3sqrt(5)` units

`PB = sqrt((4 - 2)^2 + (-3 - 1)^2`

= `sqrt(2^2 + (-4)^2`

= `sqrt(4 + 16)`

= `sqrt(20)`

= `2sqrt(5)` units

Point P(2, 1) satisfies the equation x – 3y = –1, proving it lies on the line.

The lengths are `PA = 3sqrt(5)` units and `PB = 2sqrt(5)` units.