Answer $C$
$\because $ We know that the diagonals of the $RECTANGLE$ bisect each other.
So, equation of the diagonal must be satisfied by the mid-points of the diagonal.
Now, mid-points of diagonal can be calculated by the mid-point formula as:
$\bigg (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\bigg ) =\bigg ( \frac{1+5}{2}, \frac{3+1}{2}\bigg ) = (3, 2)$
The equation of the diagonal is given as:
$$ y = 2x + c$$
Substitute the values of $x$, and $y$ coordinates, we get:
$$2 = 2 \times 3 + c \Rightarrow \color {blue}{c = -4}$$
$\therefore \;C$ is the correct option.