Question

In what ratio does the line x − y = 2 divides the line segment joining the points (3, −1) and (8, 9)? Also, find the coordinates of the point of division.

Answer 1

Let A = (3, −1) and B = (8, 9).

Parametrize the point P on AB by t (0 ≤ t ≤ 1):

P = A + t(B − A)

= (3 + 5t, −1 + 10t)

P lies on the line x − y = 2, so

(3 + 5t) − (−1 + 10t) = 2

4 − 5t = 2

t = `2/5`

Thus P = `(3 + 5 xx 2/5, −1 + 10 xx 2/5)` = (5, 3)

Since t = `"AP"/"AB" = 2/5`, the ratio AP : PB = 2 : 3

If P divides AB, A (x₁, y₁), B (x₂, y₂) internally in the ratio m : n (that is AP : PB = m : n),

Then P = `((nx_1 + mx_2)/(m + n), (ny_1 + my_2)/(m + n))`

Equivalently, taking the ratio k : 1 gives;

x = `(kx_2 + x_1) / (k + 1)`,

y = `(ky_2 + y_1) / (k + 1)`

Hence,

Ratio = 2 : 3, and Point of division = (5, 3)