13.2.5* Consider all line segments of length 4, one end of which lies on the line $y=x$, and the other end on the line $y=2x$. Find: the equation of the locus of the midpoints of these line segments.
Let $A(a, 2a)$ be any point on the line $y=2x$, and $B(b, b)$ be any point on the line $y=x$. The midpoint $M(x, y)$ of the line segment $AB$ is such that $x=\frac{a+b}{2}, y=\frac{2a+b}{2}$.
(1) From the length of line segment $AB$ being 4, we get $(a-b)^{2}+(2a-b)^{2}=4^{2}$.
(2) From (1), we have $a=2(y-x), b=2(2x-y)$. Substituting into (2) yields $[2(y-x)-2(2x-y)]^{2}+[4(y-x)-2(2x-y)]^{2}=16$, which simplifies to $25x^{2}-36xy+13y^{2}=4$.