Mutually tangent line

Calculus Level 4

A straight line is drawn in a plane such that it is tangent to both curves \(y=\sqrt {x}\) and \(y=-\frac {1} {x} \). If the area of the region bound by this line, the \(x\)-axis and the curve \(y=\sqrt {x} \) is equal to \(\frac {a} {b} \) where \(a\) and \(b\) are coprime positive integers, find the value of \(a+b\).

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