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)
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