The length of the line segment joining the two intersection points of the curves \(y = 4970 - |x|\) and \(y = x ^ 2\) is

The segment length is 140. Writing $u=|x|$, the intersection equation becomes $u^2+u=4970=70\times71$, so the two points have $x=-70$ and $x=70$ and the same $y$-coordinate. The value 70 measures the horizontal distance from the vertical axis to just one point.