Question

If the points A (1,2) , O (0,0) and C (a,b) are collinear , then find  a : b.

 

Answer 1

For the three points `(x_1,y_1) , (x_2 , y_2) " and " (x_3,y_3)` to be collinear we need to have area enclosed between the points equal to zero.
Here, points `(x_1,y_1) , (x_2 , y_2) " and " (x_3,y_3)`   are

\[A\left( 1, 2 \right), O\left( 0, 0 \right) \text{ and }  C\left( a, b \right)\]

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