(i) Let m:n be the ratio in which the line segment joining (3,4) and (-2,1) is divided by the Y axis.

Since the line meets Y axis, its x co-ordinate is zero.

Here x₁ = 3, y₁ = 4

x₂ = -2, y₂ = 1

By section formula, x = (mx₂+nx₁)/(m+n)

0 = (m×-2+n×3)/(m+n)

0 = (-2m+3n)/( m+n)

0 = -2m+3n

2m = 3n

m/n = 3/2

Hence the ration m:n is 3:2.

(ii)Let the line x-y-2 = 0 divide the line segment joining the points (3,-1) and (8,9) in the ratio m:n at the point P(x,y)

Here x₁ = 3, y₁ = -1

x₂ = 8 y₂ = 9

By section formula, x = (mx₂+nx₁)/(m+n)

x = (m×8+n×3)/(m+n)

x = (8m+3n)/( m+n) …. (i)

By Section formula y = (my₂+ny₁)/(m+n)

y = (m×9+n×-1)/(m+n)

y = (9m-n)/(m+n) …. (ii)

Since the point P(x,y) lies on the line x-y-2 = 0,

eqn (i) and (ii) will satify the equation x-y-2 = 0 …(iii)

Substitute (i) and (ii) in (iii)

[(8m+3n)/( m+n)] – [(9m-n)/(m+n)]-2 = 0

[(8m+3n)/( m+n)] – [(9m-n)/(m+n)]-[2(m+n)/(m+n)] = 0

8m+3n-(9m-n)-2(m+n) = 0

8m+3n-9m+n-2m-2n = 0

-3m+2n = 0

-3m = -2n

m/n = -2/-3 = 2/3

Hence the ratio m:n is 2:3.

Substitute m and n in (i)

x = (8m+3n)/( m+n)

x = (8×2+3×3)/(2+3)

x = (16+9)/5

x = 25/5 = 5

Substitute m and n in (ii)

y = (9m-n)/(m+n)

y = (9×2-3)/(2+3)

y = (18-3)/5

y = 15/5 = 3

Hence the co-ordinates of P are (5,3).