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

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

Solution:

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

$\left|\begin{array}{ccc}3 & -2 & 1 \\ x & 2 & 1 \\ 8 & 8 & 1\end{array}\right|=0$

$\Delta=\left|\begin{array}{ccc}3 & -2 & 1 \\ x & 2 & 1 \\ 8 & 8 & 1\end{array}\right|$

$=\left|\begin{array}{ccc}3 & -2 & 1 \\ x-3 & 4 & 0 \\ 8 & 8 & 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ]

$=\left|\begin{array}{ccc}3 & -2 & 1 \\ x-3 & 4 & 0 \\ 5 & 10 & 0\end{array}\right| \quad\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$=\left|\begin{array}{cc}x-3 & 4 \\ 5 & 10\end{array}\right|$

$=10 x-30-20$

$\Delta=10 x-50$

$\Delta=0 \quad$ [Given]

$\Rightarrow 10 x-50=0$

$\Rightarrow 10 x=50$

$\Rightarrow x=5$