Know your identities

Geometry Level 2

An isosceles right triangle \(ABC\) has legs \(AC=BC=10\). A point \(D\) is chosen on \(AC\) such that \(\tan{\angle{ABD}}=\frac{1}{5}\). Then length of \(AD\) is \(\frac{a}{b}\) for relatively prime \(a\) and \(b\). Find \(a-b\).

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