For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$ Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.
Source: 1999 Romania NMO IX p2
Tags: algebra, inequalities
For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$ Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.