Question

If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.

Answer 1

If the points (3, −2), (x, 2) and (8, 8) are collinear, then

$$\begin{gathered} \left|\begin{array}{ccc}3& -2& 1\\ x& 2& 1\\ 8& 8& 1\end{array}\right|=0 \\ ∆=\left|\begin{array}{ccc}3& -2& 1\\ x& 2& 1\\ 8& 8& 1\end{array}\right| \\ =\left|\begin{array}{ccc}3& -2& 1\\ x-3& 4& 0\\ 8& 8& 1\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_2-R_1\right] \\ =\left|\begin{array}{ccc}3& -2& 1\\ x-3& 4& 0\\ 5& 10& 0\end{array}\right|\left[\mathrm{Applying}{R}_3\to R_3-R_1\right] \\ =\left|\begin{array}{cc}x-3& 4\\ 5& 10\end{array}\right| \\ =10x-30-20 \\ ∆=10x-50 \\ ∆=0\left[\mathrm{Given}\right] \\ \Rightarrow 10x-50=0 \\ \Rightarrow 10x=50 \\ \Rightarrow x=5 \end{gathered}$$