If the points \((a, b), (c, d)\) and \((a + c, b+ d)\) are collinear, then
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(ad = bc\)
Three points are collinear exactly when the triangle they form has zero area:
\( \displaystyle \begin{vmatrix} a & b & 1\\ c & d & 1\\ a+c & b+d & 1\end{vmatrix} = 0 \)
Applying \(R_3 \rightarrow R_3 - R_1 - R_2\):
\( \displaystyle \begin{vmatrix} a & b & 1\\ c & d & 1\\ 0 & 0 & -1\end{vmatrix} = 0 \)
Expanding along \(R_3\):
\(-1(ad - bc) = 0\)
\(ad = bc\)