IF POINTS ( 2,0 ) ( 0, 5) AND ( X, Y ) ARE COLLINEAR THEN SHOW THAT X/2 + - Maths - Determinants

Question

IF POINTS ( 2,0 ) ( 0, 5) AND ( X, Y ) ARE COLLINEAR THEN SHOW
THAT X/2 + Y/5 = 1

Ankush Jain · Accepted Answer

Let A (2, 0), B (0, 5) and C (X, 4) be the given three points.

So, for A, B, C are collinear

Area of ΔABC = 0

$\Leftrightarrow | \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0 \Rightarrow | \begin{matrix} 2 & 0 & 1 \\ 0 & 5 & 1 \\ x & y & 1 \end{matrix} | = 0 \Rightarrow | \begin{matrix} 2 & 0 & 1 \\ - 2 & 5 & 0 \\ x - 2 & y - 0 & 0 \end{matrix} | = 0 [{Applying R}_{2} \to R_{2} - R_{1} {and R}_{3} \to R_{3} - R_{1}] \Rightarrow 1 | \begin{matrix} - 2 & 5 \\ x - 2 & y \end{matrix} | = 0 [Expanding {along C}_{3}]$

$\Rightarrow - 2 y - 5 (x - 2) = 0 \Rightarrow - 2 y - 5 x + 10 = 0 \Rightarrow 5 x + 2 y = 10 \Rightarrow \frac{x}{2} + \frac{4}{5} = 1 [On dividing by 10]$

Hence proved

2

IF POINTS ( 2,0 ) ( 0, 5) AND ( X, Y ) ARE COLLINEAR THEN SHOW THAT X/2 + Y/5 = 1