Question

Show that the following points are collinear:

(i) A(2, −2), B(−3, 8) and C(−1, 4)
(ii) A(−5, 1), B(5, 5) and C(10, 7)
(iii) A(5, 1), B(1, −1) and C(11, 4)
(iv) A(8, 1), B(3, −4) and C(2, −5)

Answer 1

(i)
Let A(x₁ = 2, y₁ = −2), B(x₂ = −3, y₂ = 8) and C(x₃ = −1, y₃ = 4) be the given points. Now
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =2\left(8-4\right)+\left(-3\right)\left(4+2\right)+\left(-1\right)\left(-2-8\right) \\ =8-18+10 \\ =0 \end{gathered}$$
Hence, the given points are collinear.

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

(iii)
Let A(x₁ = 5, y₁ = 1), B(x₂ = 1, y₂ = −1) and C(x₃ = 11, y₃ = 4) be the given points. Now
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =5\left(-1-4\right)+1\left(4-1\right)+11\left(1+1\right) \\ =-25+3+22 \\ =0 \end{gathered}$$
Hence, the given points are collinear.

(iv)
Let A(x₁ = 8, y₁ = 1), B(x₂ = 3, y₂ = −4) and C(x₃ = 2, y₃ = −5) be the given points. Now
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \\ =8\left(-4+5\right)+3\left(-5-1\right)+2\left(1+4\right) \\ =8-18+10 \\ =0 \end{gathered}$$
Hence, the given points are collinear.