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 (1/h²) + (1/r²)?

Correct option is A

The volume of the cone is given by:

​$$V = \frac{1}{3} \pi r^2 h$$​​

The curved surface area (CSA) of the cone is given by:

​$$CSA = \pi r l$$​​

We also know that the slant height l is related to h and r by the equation:

​$$l^2 = r^2 + h^2$$​​

Since the numerical values of volume and curved surface area are equal, we set them equal:

​$$\frac{1}{3} \pi r^2 h = \pi r l$$​​

​$$\frac{1}{3} r h = l$$​​

Squaring both sides gives:

​$$\left( \frac{1}{3} r h \right)^2 = r^2 + h^2\\\ \\\frac{r^2 h^2}{9} = r^2 + h^2\\\ \\r^2 h^2 = 9(r^2 + h^2)\\\ \\\frac{1}{h^2} + \frac{1}{r^2} = \frac{1}{9}$$​​

Thus, the value of$$\frac{1}{h^2} + \frac{1}{r^2} = \frac{1}{9}$$​.