(i) Let the ratio that the point (5,4) divide the line segment joining the points (2,1) and (7,6) be m:n,

Here x₁ = 2 , y₁ = 1 , x₂ = 7, y₂ = 6, x = 5, y = 4

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

5 = (m×7+n×2)/(m+n)

5 = (7m+2n)/(m+n)

5(m+n) = 7m+2n

5m+5n = 7m+2n

5m-7m = 2n-5n

-2m = -3n

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

Hence the ratio m:n is 3:2.

(ii) Let the ratio that the point (-4,b) divide the line segment joining the points (2,-2) and (-14,6) be m:n,

Here x₁ = 2 , y₁ = -2 , x₂ = -14, y₂ = 6, x = -4, y = b

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

-4 = (m×-14+n×2)/(m+n)

-4= (-14m+2n)/(m+n)

-4(m+n) = -14m+2n

-4m-4n = -14m+2n

-4m+14m = 2n+4n

10m = 6n

m/n = 6/10 = 3/5

Hence the ratio m:n is 3:5.

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

b= (3×6+5×-2)/(3+5)

b = (18-10)/8

b = 8/8

b = 1

Hence the value of b is 1 and the ratio m:n is 3:5.