2015

Abstract

Let the endpoints of the segment have coordinates $(a,\frac{1}{a})$ and $(b,\frac{8}{b}).$ Then the middle point has coordinates $(\frac{a+b}{2},\frac{1}{2a}+\frac{4}{b}).$ If this point lies on the hyperbola $y=\frac{1}{x}$ then $\frac{a+b}{2}(\frac{1}{2a}+\frac{4}{b})=1,$ whence $\frac{1}{4}+\frac{b}{4a}+2+\frac{2a}{b}=1,$ $\frac{b}{4a}+\frac{2a}{b}=-\frac{5}{4},$ $\left( \frac{b}{a}\right) ^2+5\frac{b}{a}+8=0.$ But the quadratic equation $x^2+5x+8=0$ has no real roots, a contradiction.