Question

If the points $A(1,2),B(0,0)$ and $C(a,b)$ are collinear, then what is the relation between $a$ and $b?$

Answer 1

Given,

Points $A,B$ and $C$ are collinear

$\therefore$ Area of $△ABC=0$

We know that,

Area of a triangle formed by $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ is given by,

Area $=|\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]|$

$\therefore$ Area of $△ABC=|\frac{1}{2}\left[1\right(0-b)+0(b-2)+a(2-0\left)\right]$|

$\Rightarrow 0=|\frac{1}{2}\left[1\right(0-b)+0(b-2)+a(2-0\left)\right]$|

$\Rightarrow 0=|-b+0+2a$|

$\Rightarrow 2a-b=0$

Hence, the relation between $a$ and $b$ is $2a-b=0$.