In each of the following find the value of k, for which the points are collinear. (i) (7,-2), (5,1), (3,k) (ii) (8,1), (k,-4), (2,-5)

(i) Let the given points be A(7,-2), B(5,1) and C(3,k). These points are collinear if area (ΔABC) = 0 ⇒ x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂) = 0. Here (x₁, y₁) = (7, -2), (x₂, y₂) = (5, 1), (x₃, y₃) = (3, k) ⇒ 7(1 - k) + 5(k + 2) + 3(-2 - 1) = 0 ⇒ 7 - 7k + 5k + 10 - 9 = 0 ⇒ 8 - 2k = 0 ⇒ 2k = 8 ⇒ k = 4. Hence the given points are collinear for k = 4.
(ii) Let the given points be A(8,1), B(k,-4) and C(2,-5). These points are collinear if area (ΔABC) = 0 ⇒ x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂) = 0. Here (x₁, y₁) = (8, 1), (x₂, y₂) = (k, -4), (x₃, y₃) = (2, -5) ⇒ 8(-4 + 5) + k(-5 - 1) + 2(1 + 4) = 0 ⇒ 8 - 6k + 10 = 0 ⇒ -6k = -18 ⇒ k = 3. Hence the given points are collinear for k = 3.