If the area of a circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius.

Let A be the area of circle

The area A increases at some uniform rate hence where k is some positive constant (positive because area is increasing)

We have to comment on the change in perimeter P that is

Now perimeter of circle means circumference of circle which is

⇒ P = 2πr

Where r is radius of circle

Differentiate with respect to t

We cannot comment as of now because we don’t know

Let us find

Now area of circle

⇒ A = πr²

Differentiate with respect to time

But given that

Put in (i)

Hence the change in perimeter is inversely varied as change in radius