Question

Find a relation between x and y, if the points A(2, 1), B(x, y) and C(7, 5) are collinear. [CBSE 2009C]

Answer 1

Let A(x₁ = 2, y₁ = 1), B(x₂ = x, y₂ = y) and C(x₃ = 7, y₃ = 5) be the given points.
The given points are collinear if
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow 2\left(y-5\right)+x\left(5-1\right)+7\left(1-y\right)=0 \\ \Rightarrow 2y-10+4x+7-7y=0 \\ \Rightarrow 4x-5y-3=0 \end{gathered}$$
Hence, the required relation is 4x − 5y − 3 = 0.