The maximum area of the rectangle whose sides pass through the vertices of a given rectangle of sides a and b is

The maximum area of the rectangle whose sides pass through the vertices of a given rectangle of sides a and b is

1. A

2(ab)

2. B

$\frac{1}{2}\left(a+b{\right)}^{2}$

3. C

$\frac{1}{2}\left({a}^{2}+{b}^{2}\right)$

4. D

$\frac{1}{2}\left({a}^{2}-{b}^{2}\right)$

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Solution:

Area,  $A=\left(a\mathrm{sin}\theta +b\mathrm{cos}\theta \right)\left(a\mathrm{cos}\theta +b\mathrm{sin}\theta \right)$

$=ab+\frac{\left({a}^{2}+{b}^{2}\right)}{2}\mathrm{sin}2\theta$

A is maximum when sin $2\theta$  is maximum.

Therefore,  ${A}_{max}=ab+\frac{\left({a}^{2}+{b}^{2}\right)}{2}=\frac{1}{2}\left(a+b{\right)}^{2}$

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