For convenience, let $c = \frac{a}{2}, d = \frac{b}{2}$. $K = \sqrt{x^2 - c^2} c = \sqrt{x^2 - d^2} d$ $c^2 (x^2 - c^2) = d^2 (x^2 - d^2)$ But $c \neq d$. Therefore $x^2 - c^2 = d^2$ $x = \boxed{ \frac{\sqrt{a^2 + b^2} }{2} }$.