Question

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

(a) a = b
(b) a = 2b
(c) 2a = b
(d) a + b = 0

Answer 1

(c) 2a = b
The given points are A(1, 2), O(0, 0) and C(a, b).
Here, $\left(x_1=1,y_1=2\right),\left(x_2=0,y_2=0\right)\text{and}\left(x_3=a,y_3=b\right)$.
Points A, O and C are collinear.
$$\begin{gathered} \Rightarrow x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow 1\left(0-b\right)+0\left(b-2\right)+a\left(2-0\right)=0 \\ \Rightarrow -b+2a=0 \\ \Rightarrow 2a=b \end{gathered}$$