Question

Find a relation between x and y, if the points A(x, y), B(−5, 7) and C(−4, 5) are collinear. [CBSE 2015]

Answer 1

Let A(x₁ = x, y₁ = y), B(x₂ = −5, y₂ = 7) and C(x₃ = −4, y₃ = 5) be the given points.
The given points are collinear if
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow x\left(7-5\right)+\left(-5\right)\left(5-y\right)+\left(-4\right)\left(y-7\right)=0 \\ \Rightarrow 7x-5x-25+5y-4y+28=0 \\ \Rightarrow 2x+y+3=0 \end{gathered}$$
Hence, the required relation is 2x + y + 3 = 0.