If the line segment joining the points P and Q(3, -4) is bisected at the origin, then the coordinates of P are:

1. (3, -2)

2. (3, -4)

3. (-3, -4)

4. (-3, 4)

Let the coordinates of P be (x, y).

Given, the origin (0, 0) is the mid-point of PQ.

By mid-point formula,

(x, y) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get :

$\Rightarrow (0, 0) = \Big(\dfrac{x + 3}{2}, \dfrac{y + (-4)}{2}\Big) \\[1em] \Rightarrow 0 = \dfrac{x + 3}{2} \text{ and } 0 = \dfrac{y - 4}{2} \\[1em] \Rightarrow x + 3 = 0 \text{ and } y - 4 = 0 \\[1em] \Rightarrow x = -3 \text{ and } y = 4 \\[1em] \Rightarrow P = (x, y) = (-3, 4).$

Hence, Option 4 is the correct option.