Question

If the points A(x, y), B(−5, 7) and C(−4, 5) are collinear, then show that 2x + y + 3 = 0.

Answer 1

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