Question

Find the value of y for which the points A(−3, 9), B(2, y) and C(4, −5) are collinear.

Answer 1

Let A(x₁ = −3, y₁ = 9), B(x₂ = 2, y₂ = y) and C(x₃ = 4, y₃ = −5) be the given points.
The given points are collinear if
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow \left(-3\right)\left(y+5\right)+2\left(-5-9\right)+4\left(9-y\right)=0 \\ \Rightarrow -3y-15-28+36-4y=0 \\ \Rightarrow 7y=36-43 \end{gathered}$$
$\Rightarrow y=-1$