Mathematics - Sequence & Series Question with Solution | TestHub

$u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$ $=\sqrt{\frac{a^2+b^2}{2}+\frac{a^2-b^2}{2}\cos 2\theta }+\sqrt{\frac{a^2+b^2}{2}+\frac{b^2-a^2}{2}\cos 2\theta }$ $\Rightarrow u^2=a^2+b^2+2\sqrt{{\left(\frac{a^2+b^2}{2}\right)}^2-{\left(\frac{a^2-b^2}{2}\right)}^2{\cos }^{2}2\theta }$ min value of $u^2=a^2+b^2+2ab$ max value of $u^2=2\left(a^2+b^2\right)$ $\Rightarrow u_{\max }^2-u_{\min }^2=(a-b{)}^2$.