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 #1
We are given the parabola $y \,=\,ax^2$
. . It opens upward, is symmetric to the yaxis, with vertex at the origin $O$.
Select any point $P(p,ap^2)$ on the parabola.
Construct the perpendicular bisector of $OP$
. . and consider its $y$intercept, $b.$
Code:

b
◊ ♥ ◊
\
 \ P
◊  \ ♠(p,ap^2)
 \ *
◊  \ * ◊
◊  * ◊
◊  * ◊
◊  * ◊
       ◊      
O
The answer is surprising.
Can anyone explain this phenomenon?