Question

Show that the following points are collinear:

(i) A(0, 1), B(1, 2) and C(−2, −1)
(ii) A(−5, 1), B(5, 5) and C(10, 7)
(iii) P(a, b + c), Q(b, c + a) and R(c, a + b)

Answer 1

(i) A(0, 1), B(1, 2) and C(−2, −1) are the given points. Then:
(x₁ = 0, y₁ = 1), (x₂ = 1, y₂ = 2) and (x₃ =−2, y₃ = −1)
$$\begin{gathered} \therefore x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =0\left(2-\left(-1\right)\right)+1\left(-1-1\right)+\left(-2\right)\left(1-2\right) \\ =0\left(3\right)+1\left(-2\right)-2\left(-1\right) \\ =0-2+2 \\ =0 \end{gathered}$$
Hence, the given points are collinear.

(ii) A(−5, 1), B(5, 5) and C(10, 7) are the given points. Then:
(x₁ = −5, y₁ = 1), (x₂ = 5, y₂ = 5) and (x₃ = 10, y₃ = 7)
$$\begin{gathered} \therefore x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =\left(-5\right)\left(5-7\right)+5\left(7-1\right)+10\left(1-5\right) \\ =-5\left(-2\right)+5\left(6\right)+10\left(-4\right) \\ =10+30-40 \\ =0 \end{gathered}$$
Hence, the given points are collinear.

(iii) P(a, b + c), Q(b, c + a) and R(c, a + b) are the given points. Then:
(x₁ = a, y₁ = b + c), (x₂ = b, y₂ = c + a) and (x₃ = c, y₃ = a + b)
$$\begin{gathered} \therefore x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =a\left(c+a-a-b\right)+b\left(a+b-b-c\right)+c\left(b+c-c-a\right) \\ =a(c-b)+b(a-c)+c(b-a) \\ =ac-ab+ba-bc+cb-ca \\ =0 \end{gathered}$$
Hence, the given points are collinear.