<p>Let <img alt="$A$" height="13" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="13" />, <img alt="$B$" height="12" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="14" />, <img alt="$C$" height="12" src="https://latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png" width="14" />, and <img alt="$D$" height="14" src="https://latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png" width="17" /> be points on the hyperbola <img alt="$\frac{x^2}{20}- \frac{y^2}{24} = 1$" height="41" src="https://latex.artofproblemsolving.com/6/6/a/66a5d2a94050e282571f8d2585581e14afe578d2.png" width="102" /> such that <img alt="$ABCD$" height="13" src="https://latex.artofproblemsolving.com/f/9/e/f9efaf9474c5658c4089523e2aff4e11488f8603.png" width="57" /> is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than <img alt="$BD^2$" height="17" src="https://latex.artofproblemsolving.com/2/1/3/213cf75c75add1bff9211cd1a1c369cc8759b04f.png" width="40" /> for all such rhombi.</p>

Question No: 423

Let $A$ , $B$ , $C$ , and $D$ be points on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi.

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Tags: Geometry

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