If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

Let ' $\displaystyle \mathrm{r}$ ' be the radius, $\displaystyle \mathrm{C}$ the circumference and $\displaystyle A$ the area of the circle. Then, $\displaystyle \frac{\mathrm{dC}}{\mathrm{dt}}=\mathrm{k}($ Constant $\displaystyle )$, also $\displaystyle \mathrm{C}=2 \pi \mathrm{r} \Rightarrow \frac{\mathrm{dC}}{\mathrm{dt}}=2 \pi \frac{\mathrm{dr}}{\mathrm{dt}} \Rightarrow \frac{\mathrm{dr}}{\mathrm{dt}}=\frac{\mathrm{k}}{2 \pi} A=\pi \mathrm{r}^{2} \Rightarrow \frac{d A}{d t}=2 \pi \frac{d \mathrm{r}}{d t}=2 \pi \mathrm{r} \cdot \frac{\mathrm{k}}{2 \pi}=\mathrm{k} \mathrm{r}, \quad \therefore$ the rate of change of area is directly proportional to its radius