Question

If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.

Answer 1

If the points (x, −2), (5, 2), (8, 8) are collinear, then

$$\begin{gathered} \left|\begin{array}{ccc}x& -2& 1\\ 5& 2& 1\\ 8& 8& 1\end{array}\right|=0 \\ ∆=\left|\begin{array}{ccc}x& -2& 1\\ 5& 2& 1\\ 8& 8& 1\end{array}\right| \\ ∆=\left|\begin{array}{ccc}x& -2& 1\\ 5-x& 4& 0\\ 8& 8& 1\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_2-R_1\right] \\ =\left|\begin{array}{ccc}x& -2& 1\\ 5-x& 4& 0\\ 8-x& 10& 0\end{array}\right|\left[\mathrm{Applying}{R}_3\to R_3-R_1\right] \\ =\left|\begin{array}{cc}5-x& 4\\ 8-x& 10\end{array}\right| \\ =50-10x-32+4x \\ =18-6x \\ ∆=18-6x \\ ∆=0\left[\mathrm{Given}\right] \\ \Rightarrow 18-6x=0 \\ \Rightarrow x=3 \end{gathered}$$