If the points $(a_1, b_1), (a_2,b_2)$ and $(a_1+ a_2, b_1 + b_2)$ are collinear, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $a_1b_2=a_2 b_1$
$\text{Points: }(a_1,b_1),\ (a_2,b_2),\ (a_1+a_2,b_1+b_2)$
Collinear $\Longrightarrow$ slopes between consecutive points are equal:
$\displaystyle \frac{b_2-b_1}{a_2-a_1}=\frac{(b_1+b_2)-b_2}{(a_1+a_2)-a_2}$
$\displaystyle \frac{b_2-b_1}{a_2-a_1}=\frac{b_1}{a_1}$
Cross-multiplying and simplifying:
$a_1(b_2-b_1)=b_1(a_2-a_1)\;\Longrightarrow\;a_1b_2-a_1b_1=a_2b_1-a_1b_1$
$\displaystyle \;a_1b_2=a_2b_1$