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) → \(a_1 b_2=a_2 b_1\)
Three points are collinear exactly when the triangle they form has zero area, that is, when this determinant is zero:
\(\displaystyle \begin{vmatrix}a_1&b_1&1\\a_2&b_2&1\\a_1+a_2&b_1+b_2&1\end{vmatrix} = 0\)
Applying \(R_3 \to R_3 - R_1 - R_2\) (this does not change the determinant):
\(\displaystyle \begin{vmatrix}a_1&b_1&1\\a_2&b_2&1\\0&0&-1\end{vmatrix} = 0\)
Expanding along the third row:
\(-1\,(a_1b_2 - a_2b_1) = 0\)
\(a_1b_2 = a_2b_1\)