For the parabola $y^{2}=2 p x$, tangents are drawn from point $P$ in the plane. The direction determinants of these, $m_{1}$ and $m_{2}$, satisfy the equation

$$
m_{1} m_{2}^{2}+m_{1}^{2} m_{2}=k
$$

where $k$ is a given constant. What is the geometric locus of point $P$?
Let $M_{1}\left(x_{1}, y_{1}\right), M_{2}\left(x_{2}, y_{2}\right)$ be two points on the parabola $y^{2}=2 p x$ whose tangents have slopes $m_{1}$ and $m_{2}$, respectively, satisfying the equation

$$
m_{1} m_{2}^{2}+m_{1}^{2} m_{2}=m_{1} m_{2}\left(m_{1}+m_{2}\right)=k \ldots
$$

For the point $M_{1}$, $m_{1}=\frac{p}{y_{1}}$, and for $M_{2}$, $m_{2}=\frac{p}{y_{2}}$. Substituting these into 1):

$$
\frac{p^{3}\left(y_{1}+y_{2}\right)}{y_{1}^{2} y_{2}^{2}}=k \ldots
$$

In problem 1446, we saw that the tangents drawn at points $M_{1}$ and $M_{2}$ of the parabola intersect at a point $P(x, y)$ such that

$$
2 y=y_{1}+y_{2} \quad \text { and } \quad x^{2}=x_{1} x_{2} \ldots
$$

Thus, from 2):

$$
\frac{p^{3}\left(y_{1}+y_{2}\right)}{2 p x_{1} \cdot 2 p x_{2}}=\frac{2 p^{3} y}{4 p^{2} x^{2}}=k \quad \text { and thus } \quad x^{2}=\frac{p}{2 k} y \ldots
$$

This means that the point $P$, from which tangents with slopes satisfying equation 1) can be drawn, lies on the parabola 4); its axis is the $Y$-axis, its vertex is the origin, and its parameter is $\frac{p}{4 k}$.

If $p>0$, then the parabola $y^{2}=2 p x$ lies on the $X$-axis, and if $k>0$, the parabola $x^{2}=\frac{p}{2 k} y$ lies on the positive side of the $Y$-axis.

![](https://cdn.mathpix.com/cropped/2024_05_02_9df53853ed530abac89eg-1.jpg?height=379&width=442&top_left_y=1220&top_left_x=836)

The latter parabola has points that lie within the former. The coordinates of the common points of the two parabolas satisfy the system of equations

$$
y^{2}=2 p x, \quad x^{2}=\frac{p}{2 k} y
$$

Eliminate $y$: substitute $y=\frac{2 k}{p} x^{2}$ into the other equation; the solutions of the resulting $\frac{4 k^{2}}{p^{2}} x^{4}-2 p x=0$ are: $x_{1}=0, x_{2}=\frac{p}{\sqrt[3]{2 k^{2}}}$. (The other two roots are complex numbers.)

Thus, the common points of the two curves correspond to the abscissas $x_{1}$ and $x_{2}$. The points of the parabola 4) that lie between $x_{1}$ and $x_{2}$ cannot have tangents drawn to the given parabola. Therefore, these points do not belong to the geometric locus of point $P$.

The geometric locus of point $P$ consists of the parts of the parabola 4) for which

$$
x\frac{p}{\sqrt[3]{2 k^{2}}}
$$

Imre Máté (Cistercian St. Stephen's Grammar School, 8th grade, Székesfehérvár)