Show that the following points are collinear by determinant metho… - Vidyadip

Question

Show that the following points are collinear by determinant method.
$\mathrm{A}(2,5), \mathrm{B}(5,7), \mathrm{C}(8,9)$

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Answer

Given $\mathrm{A} \equiv\left(x_1, y_1\right)=(2,5)$,
$
\mathrm{B} \equiv\left(x_2, y_2\right) \equiv(5,7), \mathrm{C} \equiv\left(x_3, y_3\right) \equiv(8,9)
$
If $\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|=0$ then, A, B, C are collinear
$
\begin{aligned}
\therefore & \left|\begin{array}{lll}
2 & 5 & 1 \\
5 & 7 & 1 \\
8 & 9 & 1
\end{array}\right|=2(7-9)-5(5-8)+1(45-56) \\
& =-4+15-11=-15+15=0 \\
& \therefore \mathrm{A}, \mathrm{B}, \mathrm{C} \text { are collinear. }
\end{aligned}
$
$\therefore \mathrm{A}, \mathrm{B}, \mathrm{C}$ are collinear.

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