Question

The maximum area of the rectangle that can be inscribed in the circle given by $x=3+5\cos \theta ,y=1+5\sin \theta$ in sq. units is

• A. $25$
• B. $50$
• C. $100$
• D. $50\sqrt{2}$

Answer 1

Answer: (B)

$50$

The radius of the given circle is $5$.
Let $a,b$ be the length and the breadth of the rectangle.
The diagonals of the rectangle are the diameters of the circle.
Hence,
$\sqrt{a^2+b^2}=10$
The area of the rectangle $=ab$
We have,
$(a-b{)}^2\ge 0$
$\Rightarrow (a^2+b^2)-2ab\ge 0$
$\Rightarrow 100-2\times Area\ge 0$
$Area\le 50$.
The area will be maximum when the rectangle has length and breadth equal i.e. it is a square.
Hence, the maximum area is $50$ sq.units