Question

Find the value of $\lambda$ so that the points (1, −5), (−4, 5) and $\left(\lambda ,7\right)$ are collinear.

Answer 1

If the points (1, −5), (−4, 5) and $\left(\lambda ,7\right)$ are collinear, then

$$\begin{gathered} \left|\begin{array}{ccc}1& -5& 1\\ -4& 5& 1\\ \lambda & 7& 1\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}1& -5& 1\\ -5& 10& 0\\ \lambda & 7& 1\end{array}\right|=0\left[\mathrm{Applying}{R}_2\to R_2-R_1\right] \\ \Rightarrow \left|\begin{array}{ccc}1& -5& 1\\ -5& 10& 0\\ \lambda -1& 12& 0\end{array}\right|=0\left[\mathrm{Applying}{R}_3\to R_3-R_1\right] \\ \Rightarrow ∆=\left|\begin{array}{cc}-5& 10\\ \lambda -1& 12\end{array}\right|=0 \\ \Rightarrow -60-10\left(\lambda -1\right)=0 \\ \Rightarrow -60-10\lambda +10=0 \\ \Rightarrow -10\lambda =50 \\ \Rightarrow \lambda =-5 \end{gathered}$$