The maximum area of the rectangle whose sides pass through the vertices of a given rectangle of sides `aa n db` is `2(a b)` (b) `1/2(a+b)^2` `1/2(a^2+b^2)` (d) `non eoft h e s e`

Options

A. 2(ab)
B. `(1)/(2)(a+b)^(2)`
C. `(1)/(2)(a+b)^(2)`
D. none of these

Answer

2

Explanation

area `A=(a sin theta +b cos theta)(a cos theta + b sin theta)`
`=ab(sin^(2)theta+cos^(2)theta)+(a^(2)+b^(2))sin theta cos theta`
`=ab+(a^(2)+b^(2))/(2)sin 2 theta`
A Si maximum when sin 2 `theta` is maximum .Therefore
`A_(max)=ab+(a^(2)+b^(2))/(2)=1/2(a+b)^(2)`