Heron kills it

Geometry Level 5

A triangle with sides of \(\sqrt{a^2+16b^2}\), \(\sqrt{9a^2+4b^2}\), and \(2\sqrt{a^2+b^2}\) such that \(a\) and \(b\) are positive integers with area \(30\) is given. Find the number of ordered pairs \((a,b)\) that satisfy this restriction.

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