Question

Let $ABCD$ be a rectangle whose sides are given as $a$ and $b.$ A rectangle $PQRS$ whose area is $K$ sq. units is shown in figure. Then

• A. area $K$ is maximum when $\theta =\frac{\pi }{3}$
• B. area $K$ is maximum when $\theta =\frac{\pi }{4}$
• C. maximum value of area $K$ is $\frac{1}{4}(a+b{)}^2$
• D. maximum value of area $K$ is $\frac{1}{2}(a+b{)}^2$

Answer 1

Answer: (B), (D)

maximum value of area $K$ is $\frac{1}{2}(a+b{)}^2$


$BQ=a\cos \theta ,RB=b\sin \theta$
$\Rightarrow RQ=a\cos \theta +b\sin \theta$
$CQ=a\sin \theta ,CP=b\cos \theta$
$\Rightarrow PQ=a\sin \theta +b\cos \theta$

$K=$ area of $PQRS=RQ\times PQ$
$=(b\sin \theta +a\cos \theta )(a\sin \theta +b\cos \theta )$
$=ab{\sin }^2\theta +(a^2+b^2)\sin \theta \cos \theta +ab{\cos }^2\theta$
$=ab+\frac{a^2+b^2}{2}\sin 2\theta$
$K$ is maximum when $\sin 2\theta =1\Rightarrow \theta =\frac{\pi }{4}$
$K_{max}=ab+\frac{a^2+b^2}{2}=\frac{1}{2}(a+b{)}^2$