If  $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$  then the difference between the maximum and minimum value of  $u^2$ is given by

$$\begin{gathered} u^2=a^2+b^2+2\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }\times \sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta } \\ =a^2+b^2+2\sqrt{{\sin }^2\theta .{\cos }^2\theta (a^4+b^4)+a^2{b}^2({\sin }^4\theta +{\cos }^4\theta )} \\ =a^2+b^2+2\sqrt{a^2{b}^2+{(a^2-b^2)}^2{\sin }^2\theta .{\cos }^2\theta } \\ =a^2+b^2+2\sqrt{a^2{b}^2+\frac{{(a^2-b^2)}^2}{4}{\sin }^{2}2\theta } \end{gathered}$$