Question

Rectangles are inscribed in a circle of radius $r$. The dimensions of the rectangle which has the maximum area, are

• A. $r,r$
• B. $2r,2r$
• C. $\sqrt{2}r,\sqrt{2}r$
• D. None of the above

Answer 1

The correct option is C $\sqrt{2}r,\sqrt{2}r$

Let $ABCD$ be the rectangle inscribed in a circle of radius $r$.
Let $AB=x$
and $BC=y$
Then, $x^2+y^2=4r^2$
$\Rightarrow y=\sqrt{4r^2-x^2}$ ....(i)
Area of rectangle,
$A=xy$
$=x\sqrt{4r^2-x^2}$
Let $u=A^2=x^2(4r^2-x^2)$
$\Rightarrow \frac{du}{dx}=8r^{2}x-4x^3$
Put $\frac{du}{dx}=0$ for maxima or minima.
$\therefore 4x(2r^2-x^2)=0$
$\Rightarrow x=\sqrt{2}r$
Also, $\frac{d^{2}u}{d{x}^2}=8r^2-12x^2$
$\therefore {\left(\frac{d^{2}u}{d{x}^2}\right)}_{x=\sqrt{2}r}=8r^2-24r^2<0$
$\therefore u$ and $A$ are maximum at $x=\sqrt{2}r$.
From equation (i), $y=\sqrt{2}r=x$
$\therefore$ Dimensions of the rectangle are $\sqrt{2}r$ and $\sqrt{2}r$.