Question

Assertion (A): The maximum area of the rectangle inscribed in a circle of radius 5 units is 50 square units
Reason (R): The maximum area of the rectangle inscribed in a circle is a square

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true and R is false
• D. A is false and R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

Maximum area of rectangle inscribed in a circle is $2r^2$
Here $r=5$, Area $=50$ sq.units.
Area $A=xy=\left(2rcos\theta \right)\left(2rsin\theta \right)$
$A=f\left(\theta \right)=2r^{2}sin2\theta$
$f^{\prime }\left(\theta \right)=4r^{2}cos2\theta$
For maximum or minimum,
$f^{\prime }\left(\theta \right)=0$
$\Rightarrow 2\theta =\frac{\pi }{2}$
$\Rightarrow \theta =\frac{\pi }{4}$
$f^{\prime \prime }\left(\theta \right)=-8r^2<0$
Area is maximum at $\theta =\frac{\pi }{4}$
$x=\sqrt{2}r;y=\sqrt{2}r$
Hence, the rectangle is a square.