Great question!

\(y =2x^2\) is a parabola (opening upwards, because a > 0) with its vertex at origin (0, 0)

Now, Since c > 0
\(y = - 2x^2 + 4c^2\) will be a parabola (opening downwards, because a < 0) with its vertex at y-axis and \(4c^2\) units above the x-axis i.e. (0, \(4c^2\))

In order to find out the point of intersection, we can equate the y-values;
\(2x^2 = - 2x^2 + 4c^2\)
\(4x^2 = 4c^2\)
\(x^2 = c^2\)
\(x = c\)

We also know that the parabola is a symmetric figure, which means if the line segment connecting the two points of intersection has a length of 2 then the length will be equally divided (refer to the figure below).

Hence, \(c = 1\)