# If the points $\mathrm{A}(1,2), \mathrm{O}(0,0)$ and $\mathrm{C}(a, b)$ are collinear, then(A) $a=b$(B) $a=2 b$(C) $2 a=b$(D) $a=-b$

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Given:

The points $\mathrm{A}(1,2), \mathrm{O}(0,0)$ and $\mathrm{C}(a, b)$ are collinear.

To do:

We have to choose the correct option.

Solution:

We know that,

If the points $\mathrm{A}(1,2), \mathrm{O}(0,0)$ and $\mathrm{C}(a, b)$ are collinear, then the area of triangle ABC is 0.

Area of a triangle $=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$

Therefore,

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

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

$2(0)=2a-b$

$2a=b$

Hence, the correct option is (C) $2 a=b$.

Updated on 10-Oct-2022 13:28:28