The common focus of two parabolas is the point $F(2 ; 2)$, and both pass through the points $P_{1}(4 ; 2)$ and $P_{2}(-2 ; 5)$. Determine the parameters of both parabolas.
According to the definition of a parabola, the directrix of one of the two parabolas mentioned in the problem is one of the common tangents of the circles centered at $P_{1}$ and $P_{2}$ with radii $P_{1} F$ and $P_{2} F$, respectively, while the directrix of the other parabola is the other common tangent, assuming that there are only two common tangents. It is easy to see that one of the common tangents of the circles is the $x$-axis. The equations of the circles are:

$$
(x+2)^{2}+(y-5)^{2}=25
$$

and

$$
(x-4)^{2}+(y-2)^{2}=4
$$

From these equations, we get:

$$
\begin{aligned}
& x^{2}+4 x+y^{2}-10 y+4=0 \\
& x^{2}-8 x+y^{2}-4 y+16=0
\end{aligned}
$$

The solutions to this system of equations are the coordinates of the common points of the circles. From the difference of the two equations, we get $12 x-6 y-12=0$, from which $y=2 x-2$. Substituting this into the first equation, we obtain the equation $5 x^{2}-24 x+12=0$, which gives $x_{1}=2$, or $x_{2}=2.8$ and thus $y_{1}=2 ; y_{2}=3.6$. Since the two circles intersect, there are only two common tangents, so indeed only two parabolas can satisfy the conditions of the problem. The coordinates of the points $F$ and $Q$ marked on the diagram are therefore $F(2 ; 2)$, as we already knew, and $Q(2.8 ; 3.6)$. From this, we can see that the parameter of the parabola whose directrix is the $x$-axis is 2 units. The parameter of the other parabola is the distance between $F$ and the other common tangent. Since our diagram is symmetric with respect to the line $P_{1} P_{2}$, this distance is the same as the distance between $Q$ and the $x$-axis, which is the $y$-coordinate of $Q$. Therefore, the parameter of the other parabola is 3.6 units.

Based on the work of Benedek Valkó (Fazekas M. Főv. Gyak. Gimn., III. o. t.)

![](https://cdn.mathpix.com/cropped/2024_05_02_5ee320f0b40103de6209g-1.jpg?height=466&width=621&top_left_y=1277&top_left_x=730)