`a)`

`P = ((\sqrt(a-b))/(\sqrt(a+b) + \sqrt(a-b)) + (a-b)/(\sqrt(a^2 - b^2) - a + b)) . (a^2 + b^2)/(\sqrt(a^2 - b^2)) (a>b>0)`

`=> P = [(\sqrt(a-b))/(\sqrt(a+b) + \sqrt(a-b)) + (a-b)/(\sqrt((a-b)(a+b)) - (a-b))] . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = [(\sqrt(a-b))/(\sqrt(a+b) + \sqrt(a-b)) + (a-b)/(\sqrt(a-b)(\sqrt(a+b) - \sqrt(a-b)))] . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = [(\sqrt(a-b)(\sqrt(a+b) - \sqrt(a-b)))/((\sqrt(a+b) + \sqrt(a-b))(\sqrt(a+b) - \sqrt(a-b))) + (\sqrt(a-b)(\sqrt(a+b) + \sqrt(a-b)))/((\sqrt(a+b) - \sqrt(a-b))(\sqrt(a+b) + \sqrt(a-b)))] . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = (\sqrt(a-b)(\sqrt(a+b) - \sqrt(a-b)) + \sqrt(a-b)(\sqrt(a+b) + \sqrt(a-b)))/((\sqrt(a+b) - \sqrt(a-b))(\sqrt(a+b) + \sqrt(a-b))) . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = (\sqrt(a-b)(\sqrt(a+b) - \sqrt(a-b) + \sqrt(a+b) + \sqrt(a-b)))/((\sqrt(a+b))^2 - (\sqrt(a-b))^2) . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = (\sqrt(a-b).2\sqrt(a+b))/(a+b-(a-b)) . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = (2\sqrt((a-b)(a+b)))/(2b) . (a^2 + b^2)/(\sqrt((a-b)(a+b)))`

`=> P = (a^2 + b^2)/b`

`b)`

Vì `a>b>0 => (a^2 + b^2)/b > 0 => P>0`

Ta có: `a-b=1 => a=b+1`

`=> P = ((b+1)^2+b^2)/b`

`=> P = (2b^2 + 2b +1)/b`

`=> P.b = 2b^2 + 2b +1`

`=> 2b^2 + 2b - Pb + 1 = 0`

`=> 2b^2 + (2-P)b + 1 = 0`

Để phương trình có nghiệm thì:

`\Delta >= 0`

`=> (2-P)^2 - 4.2.1 >= 0`

`=> 4-4P+P^2  - 8 >= 0`

`=> P^2 - 4P - 4 >= 0`

`=> P^2 - 4P + 4 >= 8`

`=> (P-2)^2 >= (+- 2\sqrt(2))^2`

`=>` $\left[\begin{matrix} P-2 ≥ 2\sqrt{2} \\ P-2 ≤ - 2\sqrt{2} \end{matrix}\right.$

`=>` $\left[\begin{matrix} P ≥ 2 + 2\sqrt{2} \\ P  ≤ 2 - 2\sqrt{2} (\text{Loại do P>0}) \end{matrix}\right.$

`=> P_(min) = 2+2\sqrt(2)`

Khi đó: `\Delta = 0 =>  b = (-(2-P))/(2.2) = (-2+P)/4 = (-2+2+2\sqrt(2))/4 = (\sqrt(2))/2`

`=> a = (\sqrt(2))/2 + 1 = (2+\sqrt(2))/2`

Vậy $Min$ `P = 2 + 2\sqrt(2)` khi `a = (2+\sqrt(2))/2` và `b = (\sqrt(2))/2`