Determinants and Matrices — Maths STD 11 Science — Question

$\begin{aligned} & \mathrm{M}\left(\mathrm{x}_2, \mathrm{y}_2\right)=\mathrm{M}(5,7) \\ & \mathrm{N}\left(\mathrm{x}_3 \mathrm{y}_3\right)=\mathrm{N}(8,9)\end{aligned}$

If A(ΔLMN) = 0, then the points L, M, N are collinear.

$A(\Delta L \mathrm{LN})=\frac{1}{2}\left|\begin{array}{lll}2 & 5 & 1 \\ 5 & 7 & 1 \\ 8 & 9 & 1\end{array}\right|$

$=\frac{1}{2}[2(7-9)-5(5-8)+1(45-56)]$

$=\frac{1}{2}[2(-2)-5(-3)+1(-11)]$

$=\frac{1}{2}(-4+15-11)=0$

∴ The points L, M, N are collinear.