1. The rays forming a right angle with the vertex at the coordinate origin intersect the parabola $y^{2}=2 x$ at points $X$ and $Y$. Find the geometric locus of the midpoints of the segments $X Y$.
Solution. One of the legs, for example the right $O X$, has the equation $y=t x, t>0$. Then the equation of the other leg, the line $O Y$, is $y=-\frac{1}{t} x$. For the coordinates of points $X$ and $Y$, we get $X\left(\frac{2}{t^{2}}, \frac{2}{t}\right)$ and $Y\left(2 t^{2},-2 t\right)$. The midpoint of segment $X Y$ is denoted by $M$ and its coordinates are $X\left(t^{2}+\frac{1}{t^{2}}, \frac{1}{t}-t\right)$. Let's denote
$x=t^{2}+\frac{1}{t^{2}}, y=\frac{1}{t}-t$. We observe that $y^{2}=\frac{1}{t^{2}}-2+t^{2}=x-2$. Therefore, the required geometric locus is the parabola $y^{2}=x-2$.