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Parabolic Numbers (Posted on 2017-09-18) Difficulty: 2 of 5
Consider two points on parabola y=x2, (-a,a2) and (b,b2), 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.

See The Solution Submitted by Charlie    
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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)

  Posted by JayDeeKay on 2017-09-18 13:07:49
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