Let $a$ and $b$ with $a>b>0$ be real numbers satisfying $a^2+b^2=4ab$. Find $\dfrac{\sqrt{a}}{\sqrt{b}} - \dfrac{\sqrt{b}}{\sqrt{a}}$.
+128473 sqrt a / sqrt b - sqrt b / sqrt a =
(sqrt a)^2 - (sqrt b)^2 a - b
___________________ = _______ (1)
sqrt (ab) sqrt (ab)
(a - b)^2 = a^2 - 2ab + b^2 = ( a^2 + b^2) - 2ab = ( 4ab) - 2ab = 2ab
So
(a - b)^2 = 2ab take the square root of both sides
sqrt [ (a -b)^2] = sqrt (2ab)
a - b = sqrt (2ab)
So, subbing into (1)
sqrt (2ab) sqrt 2 * sqrt (ab)
_______ = ______________ = sqrt (2)
sqrt (ab) sqrt (ab)
