# Unusual parabola problem

#### soroban

##### Well-known member

We are given the parabola $y \,=\,ax^2$
. . It opens upward, is symmetric to the y-axis, 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
Find $\displaystyle\lim_{P\to O}b$

Can anyone explain this phenomenon?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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(Hide the spoiler from the forum overview.)
The y-intercept is at twice the distance to the focal point.

This is similar to a lens.
If you have a point source at twice the focal distance of a lens, the light rays converge at the other side at twice the focal distance.

In this case we have a parabolic mirror.
When light rays start at twice the focal distance, they return to the same point.