If `u=sqrt(a^2 cos^2 theta + b^2sin^2theta)+sqrt(a^2 sin^2 theta + b^2 cos^2 theta),` then the difference between the maximum and minimum values of `u^2` is given by : (a) `(a-b)^2` (b) `2sqrt(a^2+b^2)` (c) `(a+b)^2` (d) `2(a^2+b^2)`

Options

A. `(a-b)^(2)`
B. `2sqrt(a^(2)+b^(2))`
C. `(a+b)^(2)`
D. `2(a^(2)+b^(2))`

Answer

D

Explanation

`u=sqrt(a^(2)cos^(2)theta+b^(2)sin^(2)theta)+sqrt(a^(2)sin^(2)theta+b^(2)cos^(2)theta)`
`impliesu^(2)=(a^(2)+b^(2))+2sqrt((a^(2)cos^(2)theta+b^(2)sin^(2)theta)(a^(2)sin^(2)theta+b^(2)cos^(2)theta))`
`implies{u^(2)-(a^(2)+b^(2))}^(2)`
`=4{(a^(4)+b^(4))sin^(2)theta cos^(2)theta+a^(2)b^(2)(cos^(4)theta+sin^(4)theta)}`
`implies{u^(2)-(a^(2)+b^(2))}^(2)=(a^(4)+b^(4))sin^(2)2theta+4a^(2)b^(2)(1-(1)/(2)sin^(2)2theta)`
`implies{u^(2)-(a^(2)+b^(2))}^(2)=(a^(4)+b^(4)-2a^(2)b^(2))^(2)sin^(2)2theta+4a^(2)b^(2)`
`implies{u^(2)-(a^(2)+b^(2))}^(2)=(a^(2)-b^(2))sin^(2)2 theta+4a^(2)b^(2)`
`impliesu^(4)-2u^(2)(a^(2)+b^(2))+(a^(2)-b^(2))^(2)=(a^(2)-b^(2))sin^(2)2theta`
`impliesu^(4)-2u^(2)(a^(2)+b^(2))+(a^(2)-b^(2))^(2)le(a^(2)-b^(2))^(2)[because0lesin^(2)2thetale1]`
`impliesu^(4)-2u^(2)(a^(2)+b^(2))le0`
`impliesu^(2){u^(2)-2(a^(2)+b^(2))}le0implies0leu^(2)le2(a^(2)+b^(2))`
Thus, the maximum and minimum values of `u^(2) are 2(a^(2)+b^(2))` and 0. Hence, required difference`=2(a^(2)+b^(2)).`