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Question

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


Solution

Given,
Points $A,B$ and $C$ are collinear
$\therefore$  Area of $\triangle ABC=0$

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 $=|\dfrac12[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]|$

$\therefore$ Area of $\triangle ABC=|\dfrac{1}{2}[1(0-b)+0(b-2)+a(2-0)]$|
$\Rightarrow 0=|\dfrac{1}{2}[1(0-b)+0(b-2)+a(2-0)]$|
$\Rightarrow  0= |-b+0+2a$|
$\Rightarrow 2a-b=0$

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

Mathematics

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