"Collinear" implies the three points lie on the same line. Two points will determine a linear equation if their coordinates are given. In this question, the coordinates of A and B are given, the linear equation for the line is determined. According to the condition B is the midpoint of line segment AC, we can use the coordinates formula for a midpoint between two points to determine the coordinates of C.

x_B = \dfrac{x_A+x}{2}

(1)

y_B = \dfrac{y_A+y}{2}

(2)

Substitute the x coordinates of A, B to (1)

\dfrac{\sqrt{2}+x }{2}=6

We get

x= 12-\sqrt{2}\approx 10.59

Substitute the y coordinates of A, B to (2)

\dfrac{4+y}{2}=-\sqrt{3}

Then we get

y = -2\sqrt{3} -4 \approx -7.46

So D is the correct solution.