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Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Applications of Derivatives | CBSE | Class 12 | Exercise 6.5 | Maths | NCERT
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Let PQRS be the rectangle inscribed in a given circle with centre O and radius 

Let  and  be the length and breadth of the rectangle, i.e.,  and 

In right angled triangle PQR, using Pythagoras theorem,

PQ2 + QR2 = PR2

  

 

   …..(i)

Let A be the area of the rectangle, then A =  = 

 

= 

And 

= 

 

= 

Now 

  = 0

 

  At ,   [Negative]

  At , area of rectangle is maximum.

And from eq. (i), ,

i.e., 

Therefore, the area of inscribed rectangle is maximum when it is square.

i

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