find minimum value of $u$ for $\,u = \frac{ax^2+by^2}{\sqrt{a^2x^2+b^2y^2}}$

Question

Find the minimum value of $u$ where

$x^2+y^2=1\;$ and $\;u =\displaystyle{\dfrac{ax^2+by^2}{\sqrt {a^2x^2+b^2y^2}}}$

Answer 1

HINT: Let $$E=a^2x^2+b^2y^2$$

$$(a+b)\left(a x^2+b y^2\right)=E+a b\left(x^2+y^2\right)=E+ab$$ $$\Rightarrow \frac{a x^2+b y^2}{\sqrt{E}}=\frac{\sqrt{E}}{a+b}+\frac{\left(\frac{a b}{a+b}\right)}{\sqrt{E}}$$