Consider two points on parabola y=x

^{2}, (-a,a

^{2}) and (b,b

^{2}), where a and b are distinct real numbers.

If these two points are connected by a straight line, where does that line intersect the y-axis?

Inspired by an interactive sculpture at the Museum of Mathematics, NYC.

Using the given 2 points,

the slope of the line, m = (a^2 - b^2) / (-a - b)

= b - a after some basic algebra . . . . (A)

Now using, points (b, b^2) and (0, c) where c = the required y-int, the slope is (c - b^2)/ (0 - b) = (c - b^2) / (-b) . . . . . . . . . . . .(B)

Equating (A) and (B): b - a = (b^2 - c) / b

=> __c = ab__

The line intercepts the Y-axis at (0, ab)