This week’s Riddler Classic is a neat geometry problem about

Suppose you have two distinct points anywhere on the coordinate plane. If I tell you that a parabola with a vertical line of symmetry passes through those two points, where on the plane could that parabola’s vertex be?

Here is my solution:

[Show Solution]

Since the two points can be anywhere, let’s define our coordinate system so that the origin is located at the midpoint between the two points. This way, by symmetry, we can say the two points are located at $A(-u,-v)$ and $B(u,v).$ A parabola with vertical axis of symmetry and vertex located at $(x,y)$ satisfies an equation of the form $v-y=a(u-x)^2$. Given that such a parabola passes through $A$ and $B$, the equation must be satisfied when we substitute the coordinates of $A$ and $B$:

\begin{align}

-v-y &= a(-u-x)^2 \\

v-y &= a(u-x)^2

\end{align}Eliminating $a$ from this pair of equations, we are left with a single equation that relates the coordinates of the vertex $(x,y)$ and the coordinates $u$ and $v$ that determine the locations of $A$ and $B$.

\[

y = \frac{v}{2u}\left( x + \frac{u^2}{x} \right)

\]The set of points $(x,y)$ is **a hyperbola** centered at the origin (the midpoint of $A$ and $B$), whose asymptotes are the lines $x=0$ and $y=\tfrac{v}{2u}x$. Here is a figure showing the point and the hyperbola corresponding to the locus of possible locations of the vertex.

### Geometric visualization

One way to visualize the locus of possible parabola vertex locations is to use the geometric definition of a parabola: A parabola is the set of points equidistant from a point $F$ (the focus) and a line (the directrix). Here is a diagram showing this construction in action:

As $P’$ slides along the directrix, the parabola is the set of points $P$ that are equidistant to the directrix and the point $F$.

In the original problem, we are considering parabolas with a vertical axis of symmetry, therefore the directrix must be horizontal. We can find the location of the focus by drawing circles centered at $A$ and $B$ that are tangent to the directrix, and the focus must be located at an intersection point of the circles. Finally, the vertex of the parabola is the midpoint between the focus and the directrix. As we pick different locations for the directrix, we sweep out all possible locations of the vertices.

In the diagram above, the two possible foci are at $F_1$ and $F_2$, with corresponding vertices $P_1$ and $P_2$, respectively. The associated parabola are shown in violet. As we translate the directrix vertically up and down, the points $P_1$ and $P_2$ trace out the green hyperbola. If you would like to see this for yourself, here is an interactive Geogebra visualization (you can move the directrix or the point $B$). Note: you may need to zoom/pan to see the figure.

If the above applet does not work, try this link.