1. A line $p$ passing through the origin intersects the lines $x+y-1=0$ and $x-y-1=0$ at points $A$ and $B$ respectively. Find the geometric locus of the midpoints of the segments $AB$, as the line $p$ rotates around the origin.
Solution. The equation of an arbitrary line passing through the origin has the form $y=k x$. The intersection points of this line with the lines $x+y-1=0$ and $x-y-1=0$ are $A\left(\frac{1}{1+k}, \frac{k}{1+k}\right), B\left(\frac{1}{1-k}, \frac{k}{1-k}\right)$, for $k \neq \pm 1$. Let $M(x, y)$ be the midpoint of segment $A B$. Then

$$
x=\frac{1}{1-k^{2}}, \quad y=\frac{k}{1-k^{2}}
$$

By eliminating $k$, we get $y^{2}=x(x-1)$, which represents the equation of the desired geometric locus of points, and that is a hyperbola.