A variable straight line is such that the algebraic sum of perpendicular distances from the points of intersection of the ellipse $2x^2+y^2=2$ and hyperbola $2x^2-4y^2=1$  is 0 . What is the sum of the coordinates of the fixed point through which the straight line always passes?

The ellipse and the hyperbola will intersect in four points, and it can be easily deduced  that the coordinates of these points will be
 $x=\pm \frac{3}{\sqrt{10}},y=\pm \frac{1}{\sqrt{5}}$
If the four points are represented by $(x_i,y_i),i=1,2,3,4$ , we conclude that
$\sum _{i=1}^4{x}_i=\sum _{i=1}^4{y}_i=0$ 
Now, if the variable line is represented by $ax+by+c=0$ , the (algebraic) length of the perpendicular $p_i$  from any one of the four points of intersection is
$p_i=\frac{a{x}_i+b{y}_i+c}{\sqrt{a^2+b^2}}$ 
If  $\sum _{i=1}^4{p}_i=0$, we have
 $a\sum _{i=1}^4{x}_i+b\sum _{i=1}^4{y}_i+c=0\Rightarrow c=0$
Thus, the variable line always passes through $(0,0)$ . We have $0+0=0$ , and so the correct option is (A).