If the points (x, -2), (5, 2), (8, 8) are collinear, find x using… - Vidyadip

Question

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

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Answer

The points $(k, -2), (5, 2), (8, 8)$ are collinear.
$\begin{vmatrix}\text{x}&-2&1\\5&2&1\\8&8&1\end{vmatrix}=0$
$\triangle=\begin{vmatrix}\text{x}&-2&1\\5&2&1\\8&8&1\end{vmatrix}$
$=\begin{vmatrix}\text{x}&-2&1\\5-\text{x}&4&0\\8&8&1\end{vmatrix} \ [$Applying $R_2 \rightarrow R_2 - R_1]$
$=\begin{vmatrix}\text{x}&-2&1\\5-\text{x}&4&0\\8-\text{x}&10&0\end{vmatrix}\ [$Applying $R_3 \rightarrow R_3 - R_1]$
$=\begin{vmatrix}5-\text{x}&4\\8-\text{x}&10\end{vmatrix}$
$=50-10\text{x}-32+4\text{x}$
$=18-6\text{x}=0$
$\Rightarrow\text{x}=3$

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