Step 1

Area of the circle is

$A= \pi r^2$

Differentiating w.r.t $t$ on both sides we get,

$ \large\frac{dA}{dt}= \pi.2r.\large\frac{dr}{dt}$

It is given that the area is increasing at a uniform rate.

$ \therefore \large\frac{dA}{dt}=k\: \: \Rightarrow 2\pi r.\large\frac{dr}{dt}=k$

Step 2

where $k$ is a constant $ \therefore \large\frac{dr}{dt}=\large\frac{k}{2\pi r}$

Perimeter of the circle is

$ p=2\pi r$

Differentiatily w.r.t $t$ we get

$ \large\frac{dp}{dt}=2\pi.\large\frac{dr}{dt}$

Substituting for $ \large\frac{dr}{dt}$ we get

$ \large\frac{dp}{dt}=2\pi.\large\frac{k}{2\pi r}$

$ \large\frac{dp}{dt}=\large\frac{k}{r}$

$ \Rightarrow \large\frac{dp}{dt}\: \alpha\: \large\frac{1}{r}$

Hence this proves that the perimeter varies inversely as the radius.