Question

Find the value of x for which the points $A\left(x,2\right),B\left(-3,-4\right)\mathrm{and}C\left(7,-5\right)$ are collinear. [CBSE 2015]

Answer 1

Let $A\left(x_1,y_1\right)=A\left(x,2\right),B\left(x_2,y_2\right)=B\left(-3,-4\right)\mathrm{and}C\left(x_3,y_3\right)=C\left(7,-5\right)$. So, the condition for three collinear points is
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1\mathit{-}{y}_2\right)=0 \\ \Rightarrow x\left(-4+5\right)-3\left(-5-2\right)+7\left(2+4\right)=0 \\ \Rightarrow x+21+42=0 \\ \Rightarrow x=-63 \end{gathered}$$
Hence, x = − 63.