P is the mid-point of AB and Q is the mid-point of AP. If the coordinates of A are (2, -4) and coordinates of Q are (1, -1), then the coordinates of B are:

1. (1, -1)

2. (-1, 6)

3. (-2, 6)

4. (-2, 8)

Given,

Coordinates of A = (2, -4) and Q = (1, -1)

Let the coordinates of B be (a, b).

Given that Q is the mid-point of AP and P is the mid-point of AB.

By mid-point formula,

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

Let point P be (p,q)

Since Q is the mid-point of AP, we have

$\Rightarrow (1, -1) = \Big(\dfrac{2 + p}{2}, \dfrac{-4 + q}{2}\Big) \\[1em] \Rightarrow 1 = \dfrac{2 + p}{2} \text{ and } -1 = \dfrac{-4 + q}{2} \\[1em] \Rightarrow 2 + p = 2 \text{ and } -4 + q = -2 \\[1em] \Rightarrow p = 2 - 2 \text{ and } q = -2 + 4 \\[1em] \Rightarrow p = 0 \text{ and } q = 2 .$

Hence, the coordinates of P are (0, 2).

Now, since P is the mid-point of AB. We have,

$\Rightarrow (0, 2) = \Big(\dfrac{2 + a}{2}, \dfrac{-4 + b}{2}\Big) \\[1em] \Rightarrow 0 = \dfrac{2 + a}{2} \text{ and } 2 = \dfrac{-4 + b}{2} \\[1em] \Rightarrow 2 + a = 0 \text{ and } -4 + b = 4 \\[1em] \Rightarrow a = -2 \text{ and } b = 4 + 4 \\[1em] \Rightarrow a = -2 \text{ and } b = 8.$

Therefore, the coordinates of B are (-2, 8).

Hence, Option 4 is the correct option.