Question

The maximum area of a rectangle that can be inscribed in a circle $x^2+y^2=25$ is equal to (in sq units)

Answer 1

Let $l,b$ be the length and breadth of

the rectangle, here $r=5$

we have $l^2+b^2=100\cdots \left(i\right)$

Area of rectangle, $A=lb$

Let $f=A^2=l^2{b}^2=l^2(100-l^2)$

$f^{\prime }=200l-4l^3$

$f^{\prime \prime }=200-12l^2$

$f^{\prime }=0\Rightarrow l=\sqrt{50}(l\ne 0)$

$f^{\prime \prime }\left(\sqrt{50}\right)<0$
$\Rightarrow f$ is maximum when $l=\sqrt{50}$

$\therefore b=\sqrt{50}$

Hence maximum area $=50$ sq units