The numerical values of the volume and curved surface area of a cone are equal. If '$$h$$' and '$$r$$' represent the height and base radius of the cone, then what is the value of $$\left(\frac{1}{h^2}\right) + \left(\frac{1}{r^2}\right)$$?

given that 

vol of cone = curved surface area of cone

$$\pi\times r^2\frac{h}{3}= \pi\times r\times l$$

$$r\times\frac{h}{3} = l$$ .....(1)

given that

$$\left(\frac{1}{h^2}\right) + \left(\frac{1}{r^2}\right)$$

= $$\frac{r^2 + h^2}{h^2\times r^2}$$...(2)

as we know that

$$\frac{r^2 + h^2}$$ = $$l^2$$ ....(3)

put (1) and (3) in (2)

$$\frac{r^2 + h^2}{h^2\times r^2}$$

$$\frac{l^2}{(3\times l)^2} $$

$$\frac{l^2}{9\times l^2}$$

= $$\frac{1}{9}$$