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 $

AcademicMathematicsNCERTClass 10

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) \( 2 a=b \).

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

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