If the points $(a, b), (c, d)$ and $(a + c, b+ d)$ are collinear, then

$ad = bc$

The correct answer is Option (4) → $ad = bc$

Given points:

$P(a,b),\ Q(c,d),\ R(a+c,b+d)$ are collinear.

For collinearity, the slope of $PQ$ = slope of $QR$.

Slope of $PQ = \frac{d-b}{c-a}$

Slope of $QR = \frac{(b+d)-d}{(a+c)-c} = \frac{b}{a}$

Hence,

$\frac{d-b}{c-a} = \frac{b}{a}$

Cross-multiplying,

$a(d - b) = b(c - a)$

$\Rightarrow ad - ab = bc - ab$

$\Rightarrow ad = bc$

Therefore, the required condition is $ad = bc$.