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Published in:
2017 | OriginalPaper | Chapter
2015
Authors : Volodymyr Brayman, Alexander Kukush
Published in: Undergraduate Mathematics Competitions (1995–2016)
Publisher: Springer International Publishing
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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.