If \( x+y=3 \) is tangent at \( (2,1) \) on hyperbola, intersects asymptotes at \( A \) and \( B...

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If \( x+y=3 \) is tangent at \( (2,1) \) on hyperbola, intersects asymptotes at \( A \) and \( B \) such that \( A B=8 \sqrt{2} \). If centre of hyperbola is \( (-1,-1) \), then least possible value of sum of semi-transverse axis and semi-conjugate axis is \( p \sqrt{q} \), where \( \operatorname{HCF}(p, q)=1 \) and \( \sqrt{q} \) is irrational number. Then, \( (p+q) \) is equal to
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