Two quantities $x$ and $y$ are in inverse proportion. Which relation always holds?

Two quantities $x$ and $y$ are in inverse proportion. Which relation always holds?
  1. $\dfrac{x}{y} = k$ (constant)
  2. $x + y = k$
  3. $x \cdot y = k$ (constant)
  4. $x^{2} + y^{2} = k$

Solution

By definition of inverse proportion, $xy$ is constant; equivalently $y = \dfrac{k}{x}$.

Asked in: IMO

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